introduction to stochastic calculus with applications(klebaner)

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INTRODUCTIONTOSTOCHASTICCALCULUSWITHAPPLICATIONSSECONDEDITION Thispageintentionallyleftblank FimaCKlebanerMonashUniversity,AustraliaImperialCollegePress PublishedbyImperialCollegePress57SheltonStreetCoventGardenLondonWC2H9HEDistributedbyWorldScientificPublishingCo.Pte.Ltd.5TohTuckLink,Singapore596224USAoffice:27WarrenStreet,Suite401-402,Hackensack,NJ07601UKoffice:57SheltonStreet,CoventGarden,LondonWC2H9HEBritishLibraryCataloguing-in-PublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibrary.INTRODUCTIONTOSTOCHASTICCALCULUSWITHAPPLICATIONS(SecondEdition)Copyright©2005byImperialCollegePressAllrightsreserved.Thisbook,orpartsthereof,maynotbereproducedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorageandretrievalsystemnowknownortobeinvented,withoutwrittenpermissionfromthePublisher.Forphotocopyingofmaterialinthisvolume,pleasepayacopyingfeethroughtheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,USA.Inthiscasepermissiontophotocopyisnotrequiredfromthepublisher.ISBN1-86094-555-4ISBN1-86094-566-X(pbk)PrintedinSingapore. PrefacePrefacetotheSecondEditionThesecondeditionisrevised,expandedandenhanced.ThisisnowamorecompletetextinStochasticCalculus,frombothatheoreticalandanappli-cationspointofview.Changescameabout,asaresultofusingthisbookforteachingcoursesinStochasticCalculusandFinancialMathematicsoveranumberofyears.Manytopicsareexpandedwithmoreworkedoutexamplesandexercises.Solutionstoselectedexercisesareincluded.Anewchapteronbondsandinterestratescontainsderivationsofthemainpricingmod-els,includingcurrentlyusedmarketmodels(BGM).Thechangeofnumerairetechniqueisdemonstratedoninterestrate,currencyandexoticoptions.ThepresentationofApplicationsinFinanceisnowmorecomprehensiveandself-contained.ThemodelsinBiologyintroducedintheneweditionincludetheage-dependentbranchingprocessandastochasticmodelforcompetitionofspecies.TheseMarkovprocessesaretreatedbyStochasticCalculustech-niquesusingsomenewrepresentations,suchasarelationbetweenPoissonandBirth-Deathprocesses.ThemathematicaltheoryoffilteringisbasedonthemethodsofStochasticCalculus.Inthenewedition,wederivestochasticequationsforanon-linearfilterfirstandobtaintheKalman-Bucyfilterasacorollary.Modelsarisinginapplicationsaretreatedrigorouslydemonstratinghowtoapplytheoreticalresultstoparticularmodels.Thisapproachmightnotmakecertainplaceseasyreading,however,byusingthisbook,thereaderwillaccomplishaworkingknowledgeofStochasticCalculus.PrefacetotheFirstEditionThisbookaimsatprovidingaconcisepresentationofStochasticCalculuswithsomeofitsapplicationsinFinance,EngineeringandScience.Duringthepasttwentyyears,therehasbeenanincreasingdemandfortoolsandmethodsofStochasticCalculusinvariousdisciplines.OneofthegreatestdemandshascomefromthegrowingareaofMathematicalFinance,whereStochasticCalculusisusedforpricingandhedgingoffinancialderivatives,v viPREFACEsuchasoptions.InEngineering,StochasticCalculusisusedinfilteringandcontroltheory.InPhysics,StochasticCalculusisusedtostudytheeffectsofrandomexcitationsonvariousphysicalphenomena.InBiology,StochasticCalculusisusedtomodeltheeffectsofstochasticvariabilityinreproductionandenvironmentonpopulations.Fromanappliedperspective,StochasticCalculuscanbelooselydescribedasafieldofMathematics,thatisconcernedwithinfinitesimalcalculusonnon-differentiablefunctions.Theneedforthiscalculuscomesfromthenecessitytoincludeunpredictablefactorsintomodelling.Thisiswhereprobabilitycomesinandtheresultisacalculusforrandomfunctionsorstochasticprocesses.Thisisamathematicaltext,thatbuildsontheoryoffunctionsandprob-abilityanddevelopsthemartingaletheory,whichishighlytechnical.Thistextisaimedatgraduallytakingthereaderfromafairlylowtechnicalleveltoasophisticatedone.Thisisachievedbymakinguseofmanysolvedexam-ples.Everyefforthasbeenmadetokeeppresentationassimpleaspossible,whilemathematicallyrigorous.Simpleproofsarepresented,butmoretechni-calproofsareleftoutandreplacedbyheuristicargumentswithreferencestoothermorecompletetexts.Thisallowsthereadertoarriveatadvancedresultssooner.Theseresultsarerequiredinapplications.Forexample,thechangeofmeasuretechniqueisneededinoptionspricing;calculationsofconditionalexpectationswithrespecttoanewfiltrationisneededinfiltering.Itturnsoutthatcompletelyunrelatedappliedproblemshavetheirsolutionsrootedinthesamemathematicalresult.Forexample,theproblemofpricinganoptionandtheproblemofoptimalfilteringofanoisysignal,bothrelyonthemartingalerepresentationpropertyofBrownianmotion.Thistextpresumeslessinitialknowledgethanmosttextsonthesubject(M´etivier(1982),DellacherieandMeyer(1982),Protter(1992),LiptserandShiryayev(1989),JacodandShiryayev(1987),KaratzasandShreve(1988),StroockandVaradhan(1979),RevuzandYor(1991),RogersandWilliams(1990)),howeveritstillpresentsafairlycompleteandmathematicallyrigoroustreatmentofStochasticCalculusforbothcontinuousprocessesandprocesseswithjumps.Abriefdescriptionofthecontentsfollows(formoredetailsseetheTableofContents).ThefirsttwochaptersdescribethebasicresultsinCalculusandProbabilityneededforfurtherdevelopment.Thesechaptershaveexamplesbutnoexercises.Somemoretechnicalresultsinthesechaptersmaybeskippedandreferredtolaterwhenneeded.InChapter3,thetwomainstochasticprocessesusedinStochasticCalculusaregiven:Brownianmotion(forcalculusofcontinuousprocesses)andPoissonprocess(forcalculusofprocesseswithjumps).IntegrationwithrespecttoBrownianmotionandcloselyrelatedprocesses(Itˆoprocesses)isintroducedinChapter4.Itallowsonetodefineastochasticdifferentialequation.Such PREFACEviiequationsariseinapplicationswhenrandomnoiseisintroducedintoordinarydifferentialequations.StochasticdifferentialequationsaretreatedinChapter5.Diffusionprocessesariseassolutionstostochasticdifferentialequations,theyarepresentedinChapter6.Asthenamesuggests,diffusionsdescribearealphysicalphenomenon,andaremetinmanyreallifeapplications.Chapter7containsinformationaboutmartingales,examplesofwhichareprovidedbyItˆoprocessesandcompensatedPoissonprocesses,introducedinearlierchap-ters.Themartingaletheoryprovidesthemaintoolsofstochasticcalculus.Theseincludeoptionalstopping,localizationandmartingalerepresentations.Theseareabstractconcepts,buttheyariseinappliedproblems,wheretheiruseisdemonstrated.Chapter8givesabriefaccountofcalculusformostgeneralprocesses,calledsemimartingales.BasicresultsincludeItˆo’sformulaandstochasticexponential.ThereaderhasalreadymettheseconceptsinBrownianmotioncalculusgiveninChapter4.Chapter9treatsPureJumpprocesses,wheretheyareanalyzedbyusingcompensators.ThechangeofmeasureisgiveninChapter10.Thistopicisimportantinoptionspric-ing,andforinferenceforstochasticprocesses.Chapters11-14aredevotedtoapplicationsofStochasticCalculus.ApplicationsinFinancearegiveninChapters11and12,stocksandcurrencyoptions(Chapter11);bonds,inter-estratesandtheiroptions(Chapter12).ApplicationsinBiologyaregiveninChapter13.Theyincludediffusionmodels,Birth-Deathprocesses,age-dependent(Bellman-Harris)branchingprocesses,andastochasticversionoftheLotka-Volterramodelforcompetitionofspecies.Chapter14givesap-plicationsinEngineeringandPhysics.Equationsforanon-linearfilterarederived,andappliedtoobtaintheKalman-Bucyfilter.Randomperturba-tionstotwo-dimensionaldifferentialequationsaregivenasanapplicationinPhysics.Exercisesareplacedattheendofeachchapter.ThistextcanbeusedforavarietyofcoursesinStochasticCalculusandFinancialMathematics.TheapplicationtoFinanceisextensiveenoughtouseitforacourseinMathematicalFinanceandforselfstudy.Thistextissuitableforadvancedundergraduatestudents,graduatestudentsaswellasresearchworkersandpractioners.AcknowledgmentsThankstoRobertLiptserandKaisHamzawhoprovidedmostvaluablecom-ments.ThankstotheEditorLenoreBettsforproofreadingthe2ndedition.Theremainingerrorsaremyown.Thankstomycolleaguesandstudentsfromuniversitiesandbanks.Thankstomyfamilyforbeingsupportiveandunderstanding.FimaC.KlebanerMonashUniversityMelbourne,2004. Thispageintentionallyleftblank ContentsPrefacev1PreliminariesFromCalculus11.1FunctionsinCalculus.......................11.2VariationofaFunction......................41.3RiemannIntegralandStieltjesIntegral..............91.4Lebesgue’sMethodofIntegration.................141.5DifferentialsandIntegrals.....................141.6Taylor’sFormulaandOtherResults...............152ConceptsofProbabilityTheory212.1DiscreteProbabilityModel....................212.2ContinuousProbabilityModel...................282.3ExpectationandLebesgueIntegral................332.4TransformsandConvergence...................372.5IndependenceandCovariance...................392.6Normal(Gaussian)Distributions.................412.7ConditionalExpectation......................432.8StochasticProcessesinContinuousTime.............473BasicStochasticProcesses553.1BrownianMotion..........................563.2PropertiesofBrownianMotionPaths..............633.3ThreeMartingalesofBrownianMotion..............653.4MarkovPropertyofBrownianMotion..............673.5HittingTimesandExitTimes...................693.6MaximumandMinimumofBrownianMotion..........713.7DistributionofHittingTimes...................733.8ReflectionPrincipleandJointDistributions...........743.9ZerosofBrownianMotion.ArcsineLaw.............75ix xPREFACE3.10SizeofIncrementsofBrownianMotion..............783.11BrownianMotioninHigherDimensions.............813.12RandomWalk............................813.13StochasticIntegralinDiscreteTime...............833.14PoissonProcess...........................863.15Exercises..............................884BrownianMotionCalculus914.1DefinitionofItˆoIntegral......................914.2ItˆoIntegralProcess.........................1004.3ItˆoIntegralandGaussianProcesses...............1034.4Itˆo’sFormulaforBrownianMotion................1054.5ItˆoProcessesandStochasticDifferentials............1084.6Itˆo’sFormulaforItˆoProcesses..................1114.7ItˆoProcessesinHigherDimensions................1174.8Exercises..............................1205StochasticDifferentialEquations1235.1DefinitionofStochasticDifferentialEquations..........1235.2StochasticExponentialandLogarithm..............1285.3SolutionstoLinearSDEs.....................1305.4ExistenceandUniquenessofStrongSolutions..........1335.5MarkovPropertyofSolutions...................1355.6WeakSolutionstoSDEs......................1365.7ConstructionofWeakSolutions..................1385.8BackwardandForwardEquations................1435.9StratanovichStochasticCalculus.................1455.10Exercises..............................1476DiffusionProcesses1496.1MartingalesandDynkin’sFormula................1496.2CalculationofExpectationsandPDEs..............1536.3TimeHomogeneousDiffusions...................1566.4ExitTimesfromanInterval....................1606.5RepresentationofSolutionsofODEs...............1656.6Explosion..............................1666.7RecurrenceandTransience....................1676.8DiffusiononanInterval......................1696.9StationaryDistributions......................1706.10Multi-DimensionalSDEs......................1736.11Exercises..............................180 PREFACExi7Martingales1837.1Definitions..............................1837.2UniformIntegrability........................1857.3MartingaleConvergence......................1877.4OptionalStopping.........................1897.5LocalizationandLocalMartingales................1957.6QuadraticVariationofMartingales................1987.7MartingaleInequalities.......................2007.8ContinuousMartingales.ChangeofTime............2027.9Exercises..............................2098CalculusForSemimartingales2118.1Semimartingales..........................2118.2PredictableProcesses........................2128.3Doob-MeyerDecomposition....................2148.4IntegralswithrespecttoSemimartingales............2158.5QuadraticVariationandCovariation...............2188.6Itˆo’sFormulaforContinuousSemimartingales..........2208.7LocalTimes.............................2228.8StochasticExponential.......................2248.9CompensatorsandSharpBracketProcess............2288.10Itˆo’sFormulaforSemimartingales................2348.11StochasticExponentialandLogarithm..............2368.12Martingale(Predictable)Representations............2378.13ElementsoftheGeneralTheory..................2408.14RandomMeasuresandCanonicalDecomposition........2448.15Exercises..............................2479PureJumpProcesses2499.1Definitions..............................2499.2PureJumpProcessFiltration...................2509.3Itˆo’sFormulaforProcessesofFiniteVariation..........2519.4CountingProcesses.........................2529.5MarkovJumpProcesses......................2599.6StochasticEquationforJumpProcesses.............2619.7ExplosionsinMarkovJumpProcesses..............2639.8Exercises..............................265 xiiPREFACE10ChangeofProbabilityMeasure26710.1ChangeofMeasureforRandomVariables............26710.2ChangeofMeasureonaGeneralSpace..............27110.3ChangeofMeasureforProcesses.................27410.4ChangeofWienerMeasure....................27910.5ChangeofMeasureforPointProcesses..............28010.6LikelihoodFunctions........................28210.7Exercises..............................28511ApplicationsinFinance:StockandFXOptions28711.1FinancialDerivativesandArbitrage...............28711.2AFiniteMarketModel......................29311.3SemimartingaleMarketModel..................29711.4DiffusionandtheBlack-ScholesModel..............30211.5ChangeofNumeraire........................31011.6Currency(FX)Options......................31211.7Asian,LookbackandBarrierOptions..............31511.8Exercises..............................31912ApplicationsinFinance:Bonds,RatesandOptions32312.1BondsandtheYieldCurve....................32312.2ModelsAdaptedtoBrownianMotion...............32512.3ModelsBasedontheSpotRate..................32612.4Merton’sModelandVasicek’sModel...............32712.5Heath-Jarrow-Morton(HJM)Model...............33112.6ForwardMeasures.BondasaNumeraire............33612.7Options,CapsandFloors.....................33912.8Brace-Gatarek-Musiela(BGM)Model..............34112.9SwapsandSwaptions........................34512.10Exercises..............................34713ApplicationsinBiology35113.1Feller’sBranchingDiffusion....................35113.2Wright-FisherDiffusion......................35413.3Birth-DeathProcesses.......................35613.4BranchingProcesses........................36013.5StochasticLotka-VolterraModel.................36613.6Exercises..............................373 PREFACExiii14ApplicationsinEngineeringandPhysics37514.1Filtering...............................37514.2RandomOscillators.........................38214.3Exercises..............................388SolutionstoSelectedExercises391References407Index413 Thispageintentionallyleftblank Chapter1PreliminariesFromCalculusStochasticcalculusdealswithfunctionsoftimet,0≤t≤T.Inthischaptersomeconceptsoftheinfinitesimalcalculususedinthesequelaregiven.1.1FunctionsinCalculusContinuousandDifferentiableFunctionsAfunctiongiscalledcontinuousatthepointt=t0iftheincrementofgoversmallintervalsissmall,∆g(t)=g(t)−g(t0)→0as∆t=t−t0→0.Ifgiscontinuousateverypointofitsdomainofdefinition,itissimplycalledcontinuous.giscalleddifferentiableatthepointt=t0ifatthatpoint∆g(t)∆g∼C∆torlim=C,∆t→0∆tthisconstantCisdenotedbyg(t).Ifgisdifferentiableateverypointofits0domain,itiscalleddifferentiable.Animportantapplicationofthederivativeisatheoremonfiniteincre-ments.Theorem1.1(MeanValueTheorem)Iffiscontinuouson[a,b]andhasaderivativeon(a,b),thenthereisc,ag(ti−1)resultinginatelescopingsum,whereallthetermsexcludingthefirstandthelastcancelout,leavingVg(t)=g(t)−g(0).2.Ifg(t)isdecreasingthen,similarly,Vg(t)=g(0)−g(t).Example1.5:Ifg(t)isdifferentiablewithcontinuousderivativeg(t),g(t)=tg(s)ds,andt|g(s)|ds<∞,then00tVg(t)=|g(s)|ds.0tiThiscanbeseenbyusingthedefinitionandthemeanvaluetheorem.g(s)ds=ti−1g(ξtii)(ti−ti−1),forsomeξi∈(ti−1,ti).Thus|g(s)ds|=|g(ξi)|(ti−ti−1),ti−1and
n
ntiVg(t)=lim|g(ti)−g(ti−1)|=lim|g(s)ds|ti−1i=1i=1
nt=sup|g(ξi)|(ti−ti−1)=|g(s)|ds.i=10ThelastequalityisduetothelastsumbeingaRiemannsumforthefinalintegral.Alternatively,theresultcanbeseenfromthedecompositionofthederivativeintothepositiveandnegativeparts,ttt+−g(t)=g(s)ds=[g(s)]ds−[g(s)]ds.000−+Noticethat[g(s)]iszerowhen[g(s)]ispositive,andtheotherwayaround.Usingthisonecanseethatthetotalvariationofgisgivenbythesumofthevariationoftheaboveintegrals.Buttheseintegralsaremonotonefunctionswiththevaluezeroatzero.Hencett+−Vg(t)=[g(s)]ds+[g(s)]ds00tt+−=([g(s)]+[g(s)])ds=|g(s)|ds.00 6CHAPTER1.PRELIMINARIESFROMCALCULUSExample1.6:(Variationofapurejumpfunction).Ifgisaregularright-continuous(c`adl`ag)functionorregularleft-continuous(c`agl`ad),andchangesonlybyjumps,
g(t)=∆g(s),0≤s≤tthenitiseasytoseefromthedefinitionthat
Vg(t)=|∆g(s)|.0≤s≤tExample1.7:Thefunctiong(t)=tsin(1/t)fort>0,andg(0)=0iscontinuouson[0,1],differentiableatallpointsexceptzero,buthasinfinitevariationonanyintervalthatincludeszero.Takethepartition1/(2πk+π/2),1/(2πk−π/2),k=1,2,....Thefollowingtheoremgivesnecessaryandsufficientconditionsforafunc-tiontohavefinitevariation.Theorem1.6(JordanDecomposition)Anyfunctiong:[0,∞)→IRoffinitevariationcanbeexpressedasthedifferenceoftwoincreasingfunctionsg(t)=a(t)−b(t).Onesuchdecompositionisgivenbya(t)=Vg(t)b(t)=Vg(t)−g(t).(1.10)Itiseasytocheckthatb(t)isincreasing,anda(t)isobviouslyincreasing.Therepresentationofafunctionoffinitevariationasdifferenceoftwoincreasingfunctionsisnotunique.Anotherdecompositionis11g(t)=(Vg(t)+g(t))−(Vg(t)−g(t)).22Thesum,thedifferenceandtheproductoffunctionsoffinitevariationarealsofunctionsoffinitevariation.Thisisalsotruefortheratiooftwofunctionsoffinitevariationprovidedthemodulusofthedenominatorislargerthanapositiveconstant.ThefollowingresultfollowsbyTheorem1.3,anditsproofiseasy.Theorem1.7Afinitevariationfunctioncanhavenomorethancountablymanydiscontinuities.Moreover,alldiscontinuitiesarejumps. 1.2.VARIATIONOFAFUNCTION7Proof:Itisenoughtoestablishtheresultformonotonefunctions,sinceafunctionoffinitevariationisadifferenceoftwomonotonefunctions.Amonotonefunctionhasleftandrightlimitsatanypoint,thereforeanydiscontinuityisajump.Thenumberofjumpsofsizegreaterorequalto1isnnomorethan(g(b)−g(a))n.Thesetofalljumppointsisaunionofthesetsofjumppointswiththesizeofthejumpsgreaterorequalto1.Sinceeachnsuchsetisfinite,thetotalnumberofjumpsisatmostcountable.
Asufficientconditionforacontinuousfunctiontobeoffinitevariationisgivenbythefollowingtheorem,theproofofwhichisoutlinedinExample1.5.Theorem1.8Ifgiscontinuous,gexistsand|g(t)|dt<∞thengisoffinitevariation.Theorem1.9(Banach)Letg(t)beacontinuousfunctionon[0,1],andde-notebys(a)thenumberoft’swithg(t)=a.Thenthevariationofgis∞s(a)da.−∞ContinuousandDiscretePartsofaFunctionLetg(t),t≥0,bearight-continuousincreasingfunction.Thenitcanhaveatmostcountablymanyjumps,moreoverthesumofthejumpsisfiniteoverfinitetimeintervals.Definethediscontinuouspartgdofgby

gd(t)=g(s)−g(s−)=∆g(s),(1.11)s≤t00,sothatforallx,y∈[a,b](IR)|f(x)−f(y)|≤K|x−y|α.(1.27)ALipschitzconditionisaH¨olderconditionwithα=1,|f(x)−f(y)|≤K|x−y|.(1.28)ItiseasytoseethataH¨oldercontinuousoforderαfunctionon[a,b]isalsoH¨oldercontinuousofanylesserorder.√Example1.12:Thefunctionf(x)=xon[0,∞)isH¨oldercontinuouswithα=1/2butisnotLipschitz,sinceitsderivativeisunboundednearzero.ToseethatitisH¨older,itisenoughtoshowthatforallx,y≥0thefollowingratioisbounded,√√|x−y|≤K.(1.29)|x−y|Itisanelementaryexercisetoestablishthatthelefthandsideisboundedbydividing√throughbyy(ify=0,thentheboundisobviouslyone),andapplyingl’H¨opital’srrule.Similarly|x|,00.(1.31)Theorem1.20(Gronwall’sinequality)Letf(t),g(t)andh(t)benon-negativeon[0,T],andforall0≤t≤Ttf(t)≤g(t)+h(s)f(s)ds.(1.32)0Thenfor0≤t≤Tttf(t)≤g(t)+h(s)g(s)exph(u)duds.(1.33)0sThisformcanbefoundforexample,inDieudonn´e(1960). 1.6.TAYLOR’SFORMULAANDOTHERRESULTS19SolutionofFirstOrderLinearDifferentialEquationsLineardifferentialequations,bydefinition,arelinearintheunknownfunctionanditsderivatives.Afirstorderlinearequation,inwhichthecoefficientofdx(t)doesnotvanish,canbewrittenintheformdtdx(t)+g(t)x(t)=k(t).(1.34)dtTheseequationsaresolvedbyusingtheIntegratingFactorMethod.TheintegratingfactoristhefunctioneG(t),whereG(t)ischosenbyG(t)=g(t).AftermultiplyingbothsidesoftheequationbyeG(t),integrating,andsolvingforx(t),wehavetx(t)=e−G(t)eG(s)k(s)ds+x(0)eG(0)−G(t).(1.35)0TheintegratingfactorG(t)isdetermineduptoaconstant,butitisclearfrom(1.35),thatthesolutionx(t)remainsthesame.FurtherResultsonFunctionsandIntegrationResultsgivenherearenotrequiredtounderstandsubsequentmaterial.SomeoftheseinvolvetheconceptsofasetofzeroLebesguemeasure.Thisisgiveninthenextchapter(seeSection2.2);anycountablesethasLebesguemeasurezero,buttherearealsouncountablesetsofzeroLebesguemeasure.ApartialconversetoTheorem1.8alsoholds(see,forexample,Saks(1964),Freedman(1983)p.209,forthefollowingresults).Theorem1.21(Lebesgue)Afinitevariationfunctiongon[a,b]isdiffer-entiablealmosteverywhereon[a,b].InwhatfollowssufficientconditionsforafunctiontobeLipschitzandnottobeLipschitzaregiven.1.Iffiscontinuouslydifferentiableonafiniteinterval[a,b],thenitisLipschitz.Indeed,sincefiscontinuouson[a,b],itisbounded,|f|≤K.Thereforeyy|f(x)−f(y)|=|f(t)dt|≤|f(t)|dt≤K|x−y|.(1.36)xx2.IffiscontinuousandpiecewisesmooththenitisLipschitz,theproofissimilartotheabove.3.ALipschitzfunctiondoesnothavetobedifferentiable,forexamplef(x)=|x|isLipschitzbutitisnotdifferentiableatzero. 20CHAPTER1.PRELIMINARIESFROMCALCULUS4.ItfollowsfromthedefinitionofaLipschitzfunction(1.28),thatifitisdifferentiable,thenitsderivativeisboundedbyK.5.ALipschitzfunctionhasfinitevariationonfiniteintervals,sinceforanypartition{xi}ofafiniteinterval[a,b],

|f(xi+1)−f(xi)|≤K(xi+1−xi)=K(b−a).(1.37)6.Asfunctionsoffinitevariationhavederivativesalmosteverywhere(withrespecttoLebesguemeasure),aLipschitzfunctionisdifferentiableal-mosteverywhere.(Notethatfunctionsoffinitevariationhavederivativeswhichareinte-grablewithrespecttoLebesguemeasure,butthefunctiondoesnothavetobeequaltotheintegralofthederivative.)7.ALipschitzfunctionmultipliedbyaconstant,andasumoftwoLipschitzfunctionsareLipschitzfunctions.TheproductoftwoboundedLipschitzfunctionsisagainaLipschitzfunction.8.IffisLipschitzon[0,N]foranyN>0butwiththeconstantKdepend-ingonN,thenitiscalledlocallyLipschitz.Forexample,x2isLipschitzon[0,N]foranyfiniteN,butitisnotLipschitzon[0,+∞),sinceitsderivativeisunbounded.9.Iffisafunctionoftwovariablesf(x,t)anditsatisfiesLipschitzcondi-tioninxforallt,0≤t≤T,withsameconstantKindependentoft,itissaidthatfsatisfiesLipschitzconditioninxuniformlyint,0≤t≤T.AnecessaryandsufficientconditionforafunctionftobeRiemannintegrablewasgivenbyLebesgue(see,forexample,Saks(1964),Freedman(1983)p.208).Theorem1.22(Lebesgue)Anecessaryandsufficientconditionforafunc-tionftobeRiemannintegrableonafiniteclosedinterval[a,b]isthatfisboundedon[a,b]andalmosteverywherecontinuouson[a,b],thatis,continu-ousatallpointsexceptpossiblyonasetofLebesguemeasurezero.Remark1.6:(Thisisnotusedanywhereinthebook,anddirectedonlytoreaderswithknowledgeofFunctionalAnalysis)Continuousfunctionson[a,b]withthesupremumnormh=supx∈[a,b]|h(x)|isaBanachspace,denotedC([a,b]).ByaresultinFunctionalAnalysis,anylinearfunctionalonthisspacecanberepresentedash(x)dg(x)forsome[a,b]functiongoffinitevariation.Inthisway,theBanachspaceoffunctionsoffinitevariationon[a,b]withthenormg=Vg([a,b])canbeidentifiedwiththespaceoflinearfunctionalsonthespaceofcontinuousfunctions,inotherwords,thedualspaceofC([a,b]). Chapter2ConceptsofProbabilityTheoryInthischapterwegivefundamentaldefinitionsofprobabilisticconcepts.Sincethetheoryismoretransparentinthediscretecase,itispresentedfirst.Themostimportantconceptsnotmetinelementarycoursesarethemodelsforin-formation,itsflowandconditionalexpectation.Thisisonlyabriefdescription,andamoredetailedtreatmentcanbefoundinmanybooksonProbabilityThe-ory,forexample,Breiman(1968),Loeve(1978),Durret(1991).Conditionalexpectationwithitspropertiesiscentralforfurtherdevelopment,butsomeofthematerialinthischaptermaybetreatedasanappendix.2.1DiscreteProbabilityModelAprobabilitymodelconsistsofafilteredprobabilityspaceonwhichvariablesofinterestaredefined.Inthissectionweintroduceadiscreteprobabilitymodelbyusinganexampleofdiscretetradinginstock.FilteredProbabilitySpaceAfilteredprobabilityspaceconsistsof:asamplespaceofelementaryevents,afieldofevents,aprobabilitydefinedonthatfield,andafiltrationofincreasingsubfields.21 22CHAPTER2.CONCEPTSOFPROBABILITYTHEORYSampleSpaceConsiderasinglestockwithpriceStattimet=1,2,...T.DenotebyΩthesetofallpossiblevaluesofstockduringthesetimes.Ω={ω:ω=(S1,S2,...,ST)}.Ifweassumethatthestockpricecangoupbyafactoruanddownbyafactord,thentherelevantinformationreducestotheknowledgeofthemovementsateachtime.Ω={ω:ω=(a1,a2,...,aT)}at=uord.Tomodeluncertaintyaboutthepriceinthefuture,we“list”allpossiblefutureprices,andcallitpossiblestatesoftheworld.Theunknownfutureisjustoneofmanypossibleoutcomes,calledthetruestateoftheworld.Astimepassesmoreandmoreinformationisrevealedaboutthetruestateoftheworld.Attimet=1weknowpricesS0andS1.Thusthetruestateoftheworldliesinasmallerset,subsetofΩ,A⊂Ω.AfterobservingS1weknowwhichpricesdidnothappenattime1.ThereforeweknowthatthetruestateoftheworldisinAandnotinΩA=A¯.FieldsofEventsDefinebyFttheinformationavailabletoinvestorsattimet,whichconsistsofstockpricesbeforeandattimet.ForexamplewhenT=2,att=0wehavenoinformationaboutS1andS2,andF0={∅,Ω},allweknowisthatatruestateoftheworldisinΩ.Considerthesituationatt=1.Supposeatt=1stockwentupbyu.ThenweknowthatthetruestateoftheworldisinA,andnotinitscomplementA¯,whereA={(u,S2),S2=uord}={(u,u),(u,d)}.Thusourinformationattimet=1isF1={∅,Ω,A,A¯}.NotethatF0⊂F1,sincewedon’tforgetthepreviousinformation.AttimetinvestorsknowwhichpartofΩcontainsthetruestateoftheworld.Ftiscalledafieldoralgebraofsets.Fisafieldif1.∅,Ω∈F2.IfA∈F,andB∈FthenA∪B∈F,A∩B∈F,AB∈F. 2.1.DISCRETEPROBABILITYMODEL23Example2.1:(Examplesoffields.)Itiseasytoverifythatanyofthefollowingisafieldofsets.1.{Ω,∅}iscalledthetrivialfieldF0.2.{∅,Ω,A,A¯}iscalledthefieldgeneratedbysetA,anddenotedbyFA.Ω3.{A:A⊆Ω}thefieldofallthesubsetsofΩ.Itisdenotedby2.ApartitionofΩisacollectionofexhaustiveandmutuallyexclusivesubsets,{D1,...,Dk},suchthatDi∩Dj=∅,andDi=Ω.iThefieldgeneratedbythepartitionisthecollectionofallfiniteunionsofDj’sandtheircomplements.Thesesetsarelikethebasicbuildingblocksforthefield.IfΩisfinitethenanyfieldisgeneratedbyapartition.Ifonefieldisincludedintheother,F1⊂F2,thenanysetfromF1isalsoinF2.Inotherwords,asetfromF1iseitherasetoraunionofsetsfromthepartitiongeneratingF2.ThismeansthatthepartitionthatgeneratesF2has“finer”setsthantheonesthatgenerateF1.FiltrationAfiltrationIFisthecollectionoffields,IF={F0,F1,...,Ft,...,FT}Ft⊂Ft+1.IFisusedtomodelaflowofinformation.Astimepasses,anobserverknowsmoreandmoredetailedinformation,thatis,finerandfinerpartitionsofΩ.Intheexampleofthepriceofstock,IFdescribeshowtheinformationaboutpricesisrevealedtoinvestors.ΩExample2.2:IF={F0,FA,2},isanexampleoffiltration.StochasticProcessesIfΩisafinitesamplespace,thenafunctionXdefinedonΩattachesnumericalvaluestoeachω∈Ω.SinceΩisfinite,Xtakesonlyfinitelymanyvaluesxi,i=1,...,k.IfafieldofeventsFisspecified,thenanysetinitiscalledameasurableset.IfF=2Ω,thenanysubsetofΩismeasurable.AfunctionXonΩiscalledF-measurableorarandomvariableon(Ω,F)ifallthesets{X=xi},i=1,...,k,aremembersofF.ThismeansthatifwehavetheinformationdescribedbyF,thatis,weknowwhicheventinFhasoccurred,thenweknowwhichvalueofXhasoccurred.NotethatifF=2Ω,thenanyfunctiononΩisarandomvariable. 24CHAPTER2.CONCEPTSOFPROBABILITYTHEORYExample2.3:Considerthemodelfortradinginstock,t=1,2,whereateachtimethestockcangoupbythefactoruordownbythefactord.Ω={ω1=(u,u),ω2=(u,d),ω3=(d,u),ω4=(d,d)}.TakeA={ω1,ω2},whichisΩtheeventthatatt=1thestockgoesup.F1={∅,Ω,A,A¯},andF2=2containsall16subsetsofΩ.ConsiderthefollowingfunctionsonΩ.X(ω1)=X(ω2)=1.5,X(ω3)=X(ω4)=0.5.XisarandomvariableonF1.Indeed,theset{ω:X(ω)=21.5}={ω1,ω2}=A∈F1.Also{ω:X(ω)=0.5}=A¯∈F1.IfY(ω1)=(1.5),2Y(ω2)=0.75,Y(ω3)=0.75,andY(ω4)=0.5,thenYisnotarandomvariableonF1,itisnotF1-measurable.Indeed,{ω:Y(ω)=0.75}={ω2,ω3}∈F/1.YisF2-measurable.Definition2.1Astochasticprocessisacollectionofrandomvariables{X(t)}.Foranyfixedt,t=0,1,...,T,X(t)isarandomvariableon(Ω,FT).AstochasticprocessiscalledadaptedtofiltrationIFifforallt=0,1,...,T,X(t)isarandomvariableonFt,thatis,ifX(t)isFt-measurable.Example2.4:(Example2.3continued.)X1=X,X2=YisastochasticprocessadaptedtoIF={F1,F2}.Thisprocessrepresentsstockpricesattimetundertheassumptionthatthestockcanappreciateordepreciateby50%inaunitoftime.FieldGeneratedbyaRandomVariableLet(Ω,2Ω)beasamplespacewiththefieldofallevents,andXbearandomvariablewithvaluesxi,i=1,2,...k.ConsidersetsAi={ω:X(ω)=xi}⊆Ω.ThesesetsformapartitionofΩ,andthefieldgeneratedbythispartitioniscalledthefieldgeneratedbyX.ItisthesmallestfieldthatcontainsallthesetsoftheformAi={X=xi}anditisdenotedbyFXorσ(X).ThefieldgeneratedbyXrepresentstheinformationwecanextractaboutthetruestateωbyobservingX.Example2.5:(Example2.3continued.){ω:X(ω)=1.5}={ω1,ω2}=A,{ω:X(ω)=0.5}={ω3,ω4}=A¯.FX=F1={∅,Ω,A,A¯}.FiltrationGeneratedbyaStochasticProcessGiven(Ω,F)andastochasticprocess{X(t)}letFt=σ({Xs,0≤s≤t})bethefieldgeneratedbyrandomvariablesXs,s=0,...,t.Itisalltheinformationavailablefromtheobservationoftheprocessuptotimet.Clearly,Ft⊆Ft+1,sothatthesefieldsformafiltration.Thisfiltrationiscalledthenaturalfiltrationoftheprocess{X(t)}. 2.1.DISCRETEPROBABILITYMODEL25IfA∈Ftthenbyobservingtheprocessfrom0totweknowattimetwhetherthetruestateoftheworldisinAornot.Weillustratethisonourfinancialexample.Example2.6:TakeT=3,andassumethatateachtradingtimethestockcangoupbythefactoruordownbyd.uuuduuB¯uuddudΩ=ududduBudddddAA¯LookatthesetsgeneratedbyinformationaboutS1.ThisisapartitionofΩ,{A,A¯}.Togetherwiththeemptysetandthewholeset,thisisthefieldF1.SetsgeneratedbyinformationaboutS2areBandB¯.ThusthesetsformedbyknowledgeofS1andS2isthepartitionofΩ,consistingofallintersectionsoftheabovesets.TogetherwiththeemptysetandthewholesetthisisthefieldF2.ClearlyanysetinF1isalsoinF2,forexampleA=(A∩B)∪(A∩B¯).SimilarlyifweaddinformationaboutΩS3weobtainalltheelementarysets,ω’sandhenceallsubsetsofΩ,F3=2.InparticularwewillknowthetruestateoftheworldwhenT=3.F0⊂F1⊂F2⊂F3isthefiltrationgeneratedbythepriceprocess{St,t=1,2,3}.PredictableProcessesSupposethatafiltrationIF=(F0,F1,...,Ft,...,FT)isgiven.AprocessHtiscalledpredictable(withrespecttothisfiltration)ifforeacht,HtisFt−1-measurable,thatis,thevalueoftheprocessHattimetisdeterminedbytheinformationuptoandincludingtimet−1.Forexample,thenumberofsharesheldattimetisdeterminedonthebasisofinformationuptoandincludingtimet−1.Thusthisprocessispredictablewithrespecttothefiltrationgeneratedbythestockprices.StoppingTimesτiscalledarandomtimeifitisanon-negativerandomvariable,whichcanalsotakevalue∞on(Ω,FT).SupposethatafiltrationIF=(F0,F1,...,Ft,...,FT)isgiven.τiscalledastoppingtimewithrespecttothisfiltrationifforeacht=0,1,...,Ttheevent{τ≤t}∈Ft.(2.1) 26CHAPTER2.CONCEPTSOFPROBABILITYTHEORYThismeansthatbyobservingtheinformationcontainedinFtwecandecidewhethertheevent{τ≤t}hasorhasnotoccurred.IfthefiltrationIFisgeneratedby{St},thenbyobservingtheprocessuptotimet,S0,S1,...,St,wecandecidewhethertheevent{τ≤t}hasorhasnotoccurred.ProbabilityIfΩisafinitesamplespace,thenwecanassigntoeachoutcomeωaprobability,P(ω),thatis,thelikelihoodofitoccurring.Thisassignmentcanbearbitrary.TheonlyrequirementisthatP(ω)≥0andΣP(ω)=P(Ω)=1.Example2.7:TakeT=2inourbasicexample2.3.Ifthestockgoesupordownindependentlyofitsvalueandif,say,theprobabilitytogoupis0.4thenΩ={(u,u);(u,d);(d,u)(d,d)}P(ω)0.160.240.240.36DistributionofaRandomVariableSincearandomvariableXisafunctionfromΩtoIR,andΩisfinite,Xcantakeonlyfinitelymanyvalues,asthesetX(Ω)isfinite.Denotethesevaluesbyxi,i=1,2,...k.Thecollectionofprobabilitiespiofsets{X=xi}={ω:X(ω)=xi}iscalledtheprobabilitydistributionofX;fori=1,2,...k
pi=P(X=xi)=P(ω).ω:X(ω)=xiExpectationIfXisarandomvariableon(Ω,F)andPisaprobability,thentheexpectationofXwithrespecttoPis
EPX=X(ω)P(ω),wherethesumistakenoverallelementaryoutcomesω.ItcanbeshownthattheexpectationcanbecalculatedbyusingtheprobabilitydistributionofX,
kEPX=xiP(X=xi).i=1OfcourseiftheprobabilityPischangedtoanotherprobabilityQ,thenthesamerandomvariableXmayhaveadifferentprobabilitydistributionqi=kQ(X=xi),andadifferentexpectedvalue,EQX=i=1xiqi.WhentheprobabilityPisfixed,oritisclearfromthecontextwithrespecttowhichprobabilityPtheexpectationistaken,thenthereferencetoPisdroppedfromthenotation,andtheexpectationisdenotedsimplybyE(X)orEX. 2.1.DISCRETEPROBABILITYMODEL27ConditionalProbabilitiesandExpectationsLet(Ω,2Ω,P)beafiniteprobabilityspace,andGbeafieldgeneratedbyapartitionofΩ,{D1,...,Dk},suchthatDi∩Dj=∅,and∪iDi=Ω.RecallthatifDisaneventofpositiveprobability,P(D)>0,thentheconditionalprobabilityofaneventAgiventheeventDisdefinedbyP(A∩D)P(A|D)=.P(D)SupposethatallthesetsDiinthepartitionhaveapositiveprobability.TheconditionalprobabilityoftheeventAgiventhefieldGistherandomvariablethattakesvaluesP(A|Di)onDi,i=1,...k.LetIDdenotetheindicatorofthesetD,thatis,ID(ω)=1ifω∈DandID(ω)=1ifω∈D¯.Usingthisnotation,theconditionalprobabilitycanbewrittenas
kP(A|G)(ω)=P(A|Di)IDi(ω).(2.2)i=1Forexample,ifG={∅,Ω}isthetrivialfield,thenP(A|G)=P(A|Ω)=P(A).LetnowYbear.v.thattakesvaluesy1,...,yk,thenthesetsDi={ω:Y(ω)=yi},i=1,...k,formapartitionofΩ.IfFYdenotesthefieldgeneratedbyY,thentheconditionalprobabilitygivenFYisdenotedbyP(A|FY)=P(A|Y).Itwasassumedsofarthatallthesetsinthepartitionhaveapositiveprob-ability.Ifthepartitioncontainsasetofzeroprobability,callitN,thentheconditionalprobabilityisnotdefinedonNbytheaboveformula(2.2).Itcanbedefinedforanω∈Narbitrarily.Consequentlyanyrandomvariablewhichisdefinedby(2.2)andisdefinedarbitrarilyonNisaversionoftheconditionalprobability.AnytwoversionsonlydifferonN,whichisasetofprobabilityzero.ConditionalExpectationInthissectionletXtakevaluesx1,...,xpandA1={X=x1},...,Ap={X=xp}.LetthefieldGbegeneratedbyapartition{D1,D2,...,Dk}ofΩ.ThentheconditionalexpectationofXgivenGisdefinedby
pE(X|G)=xiP(Ai|G).i=1 28CHAPTER2.CONCEPTSOFPROBABILITYTHEORYNotethatE(X|G)isalinearcombinationofrandomvariables,sothatitisarandomvariable.ItisclearthatP(A|G)=E(IA|G),andE(X|F0)=EX,whereF0={∅,Ω}isthetrivialfield.Bythedefinitionofmeasurability,XisG-measurableifandonlyifforanyi,{X=xi}=AiisamemberofG,whichmeansthatitiseitheronepoftheDj’soraunionofsomeoftheDj’s.SinceX(ω)=i=1xiIAi(ω),akG-measurableXcanbewrittenasX(ω)=j=1xjIDj(ω),wheresomexj’smaybeequal.ItiseasytoseenowthatifXisG-measurable,thenE(X|G)=X.Notethatsincetheconditionalprobabilitiesaredefineduptoanullset,soistheconditionalexpectation.IfXandYarerandomvariablesbothtakingafinitenumberofvalues,thenE(X|Y)isdefinedasE(X|G),whereG=FYisthefieldgeneratedbytherandomvariableY.InotherwordsifXtakesvaluesx1,...,xpandYtakesvaluesy1,...,yk,andP(Y=yi)>0foralli=1,...k,thenE(X|Y)isparandomvariable,whichtakesvaluesj=1xjP(X=xj|Y=yi)ontheset{Y=yi}i=1,...k.ThesevaluesaredenotedbyE(X|Y=yi).ItisclearfromthedefinitionthatE(X|Y)isafunctionofY,
k
pE(X|Y)(ω)=E(X|FY)(ω)=xjP(X=xj|Y=yi)I{Y=yi}(ω).i=1j=12.2ContinuousProbabilityModelInthissectionwedefinesimilarprobabilisticconceptsforacontinuoussamplespace.Westartwithgeneraldefinitions.σ-FieldsAσ-fieldisafield,whichisclosedwithrespecttocountableunionsandcount-ableintersectionsofitsmembers,thatisacollectionofsubsetsofΩthatsatisfies1.∅,Ω∈F.2.A∈F⇒A¯∈F.∞∞3.A1,A2,...,An,...∈Fthenn=1An∈F(andthenalson=1An∈F).AnysubsetBofΩthatbelongstoFiscalledameasurableset. 2.2.CONTINUOUSPROBABILITYMODEL29Borelσ-fieldBorelσ-fieldisthemostimportantexampleofaσ-fieldthatisusedinthetheoryoffunctions,Lebesgueintegration,andprobability.Considertheσ-fieldBonIR(Ω=IR)generatedbytheintervals.Itisobtainedbytakingalltheintervalsfirstandthenallthesetsobtainedfromtheintervalsbyformingcountableunions,countableintersectionsandtheircomplementsareincludedintocollection,andcountableunionsandintersectionsofthesesetsareincluded,etc.Itcanbeshownthatweendupwiththesmallestσ-fieldwhichcontainsalltheintervals.Arigorousdefinitionfollows.Onecanshowthattheintersectionofσ-fieldsisagainaσ-field.Taketheintersectionofallσ-fieldscontainingthecollectionofintervals.Itisthesmallestσ-fieldcontainingtheintervals,theBorelσ-fieldonIR.InthismodelameasurablesetisasetfromB,aBorelset.ProbabilityAprobabilityPon(Ω,F)isasetfunctiononF,suchthat1.P(Ω)=1,2.IfA∈F,thenP(A¯)=1−P(A),3.Countableadditivity(σ-additivity):ifA1,A2,...,An,...∈Faremu-∞∞tuallyexclusive,thenP(n=1An)=n=1P(An).Theσ-additivitypropertyisequivalenttofiniteadditivityplusthecontinuitypropertyofprobability,whichstates:ifA1⊇A2⊇...⊇An⊇...⊇A=∩∞A∈F,thenn=1nlimP(An)=P(A).n→∞Asimilarpropertyholdsforanincreasingsequenceofevents.Howcanonedefineaprobabilityonaσ-field?Itisnothardtoseethatitisimpossibletoassignprobabilitiestoallindividualω’ssincetherearetoomanyofthemandP({ω})=0.OntheotherhanditisdifficulttoassignprobabilitiestosetsinFdirectly,sinceingeneralwedon’tevenknowwhatasetfromFlookslike.Thestandardwayistodefinetheprobabilityonafieldwhichgeneratestheσ-field,andthenextendittotheσ-field.Theorem2.2(CaratheodoryExtensionTheorem)IfasetfunctionPisdefinedonafieldF,satisfiesP(Ω)=1,P(A¯)=1−P(A),andiscountablyadditive,thenthereisauniqueextensionofPtotheσ-fieldgeneratedbyF. 30CHAPTER2.CONCEPTSOFPROBABILITYTHEORYLebesgueMeasureAsanapplicationoftheaboveTheorem2.2wedefinetheLebesguemeasureon[0,1].LetΩ=[0,1],andtakeforFtheclassofallfiniteunionsofdisjointintervalscontainedin[0,1].Itisclearlyafield.DefinetheprobabilityP(A)onFbythelengthofA.ItisnothardtoshowthatPisσ-additiveonF.ThusthereisauniqueextensionofPtoB,theBorelσ-fieldgeneratedbyF.ThisextensionistheLebesguemeasureonB.ItisalsoaprobabilityonB,sincethelengthof[0,1]isone.AnypointhasLebesguemeasurezero.Indeed,{x}=∩n(x−1/n,x+1/n).ThereforeP({x})=limn→∞2/n=0.BycountableadditivityitfollowsthatanycountablesethasLebesguemeasurezero.Inparticularthesetofrationalson[0,1]isofzeroLebesguemeasure.Thesetofirrationalson[0,1]hasLebesguemeasure1.Theterm“almosteverywhere”(for“almostallx”)meanseverywhere(forallx)except,perhaps,asetofLebesguemeasurezero.RandomVariablesArandomvariableXon(Ω,F)isameasurablefunctionfrom(Ω,F)to(IR,B),whereBistheBorelσ-fieldontheline.ThismeansthatforanyBorelsetB∈Btheset{ω:X(ω)∈B}isamemberofF.InsteadofverifyingthedefinitionforallBorelsets,itisenoughtohavethatforallrealxtheset{ω:X(ω)≤x}∈F.Insimplewords,forarandomvariablewecanassignprobabilitiestosetsoftheform{X≤x},and{a0P(|Xn−X|> )→0asn→∞.Definition2.13{Xn}convergealmostsurely(a.s.)toXifforanyωoutsideasetofzeroprobabilityXn(ω)→X(ω)asn→∞.Almostsureconvergenceimpliesconvergenceinprobability,whichinturnimpliesconvergenceindistribution.Itisalsonothardtoseethatconvergenceindistributiontoaconstantisthesameastheconvergenceinprobabilitytothesameconstant.Convergenceinprobabilityimpliesthealmostsureconvergenceonasubsequence,namely,if{Xn}convergeinprobabilitytoXthenthereisasubsequencenkthatconvergesalmostsurelytothesamelimit.Lr-convergence(convergenceinther-thmean),r≥1,isdefinedasfollows. 2.5.INDEPENDENCEANDCOVARIANCE39Definition2.14{X}convergetoXinLrifforanynE(|X|r)<∞,andnnE(|X−X|r)→0asn→∞.nUsingtheconceptofuniformintegrability,givenlaterinChapter7,conver-genceinLrisequivalenttoconvergenceinprobabilityanduniformintegrabil-ityof|X|r(seeforexample,Loeve(1978)p.164).nThefollowingresult,whichisknownasSlutskiitheorem,isfrequentlyusedinapplications.Theorem2.15IfXnconvergestoXandYnconvergestoY,thenXn+YnconvergestoX+Y,foranytypeofstochasticconvergence,exceptforconvergenceindistribution.However,ifY=0orXnandYnareindependent,thentheresultisalsotrueforconvergenceindistribution.ConvergenceofExpectationsTheorem2.16(Monotoneconvergence)IfXn≥0,andXnareincreas-ingtoalimitX,whichmaybeinfinite,thenlimn→∞EXn=EX.Theorem2.17(Fatou’slemma)IfXn≥0(orXn≥c>−∞),thenE(liminfnXn)≤liminfnEXn.Theorem2.18(DominatedConvergence)Iflimn→∞Xn=Xinproba-bilityandforalln|Xn|≤YwithEY<∞,thenlimn→∞EXn=EX.2.5IndependenceandCovarianceIndependenceTwoeventsAandBarecalledindependentifP(A∩B)=P(A)P(B).AcollectionofeventsAi,i=1,2,...iscalledindependentifforanyfinitenandanychoiceofindicesik,k=1,2,...nn!nPAik=P(Aik).k=1k=1Twoσ-fieldsarecalledindependentifforanychoiceofsetsfromeachofthem,thesesetsareindependent.TworandomvariablesXandYareindependentiftheσ-fieldstheygen-erateareindependent.Itfollowsthattheirjointdistributionisgivenbytheproductoftheirmarginaldistributions(sincethesets{X≤x}and{Y≤y}areintherespectiveσ-fields)P(X≤x,Y≤y)=P(X≤x)P(Y≤y), 40CHAPTER2.CONCEPTSOFPROBABILITYTHEORYandcanbeseenthatitisanequivalentproperty.OnecanformulatetheindependencepropertyintermsofexpectationsbywritingaboveintermsofindicatorsEI(X≤x)I(Y≤y)=EI(X≤x)EI(Y≤y).Sinceitispossibletoapproximateindicatorsbycontinuousboundedfunctions,XandYareindependentifandonlyifforanyboundedcontinuousfunctionsfandg,E(f(X)g(Y))=E(f(X))E(g(Y)).X1,X2,...,Xnarecalledindependentifforanychoiceofrandomvari-ablesXi1,Xi2,...Xiktheirjointdistributionisgivenbytheproductoftheirmarginaldistributions(alternatively,iftheσ-fieldstheygenerateareindepen-dent).CovarianceThecovarianceoftwointegrablerandomvariablesXandYisdefined,pro-videdXYisintegrable,byCov(X,Y)=E(X−EX)(Y−EY)=E(XY)−EXEY.(2.10)ThevarianceofXisthecovarianceofXwithitself,Var(X)=Cov(X,X).TheCauchy-Schwarzinequality(E|XY|)2≤E(X2)E(Y2),(2.11)assuresthatcovarianceexistsforsquareintegrablerandomvariables.Covari-anceissymmetric,Cov(X,Y)=Cov(Y,X),andislinearinbothvariables(bilinear)Cov(aX+bY,Z)=aCov(X,Z)+bCov(Y,Z).UsingthispropertywithX+Yweobtaintheformulaforthevarianceofthesum.ThefollowingpropertyofthecovarianceholdsVar(X+Y)=Var(X)+Var(Y)+2Cov(X,Y).(2.12)RandomvariablesXandYarecalleduncorrelatedifCov(X,Y)=0.ItiseasytoseefromthedefinitionsthatforindependentrandomvariablesE(XY)=EXEY,whichimpliesthattheyareuncorrelated.Theoppositeimplicationisnottrueingeneral.TheimportantexceptionistheGaussiancase. 2.6.NORMAL(GAUSSIAN)DISTRIBUTIONS41Theorem2.19IftherandomvariableshaveajointGaussiandistribution,thentheyareindependentifandonlyiftheyareuncorrelated.Definition2.20ThecovariancematrixofarandomvectorX=(X1,X2,...,Xn)isthen×nmatrixwiththeelementsCov(Xi,Xj).2.6Normal(Gaussian)DistributionsTheNormal(Gaussian)probabilitydensityisgivenby221−(x−µ)f(x;µ,σ)=√e2σ2.2πσItiscompletelyspecifiedbyitsmeanµanditsstandarddeviationσ.TheNormalfamilyN(µ,σ2)isobtainedfromtheStandardNormalDistribution,N(0,1)byalineartransformation.2X−µIfXisN(µ,σ)thenZ=isN(0,1)andX=µ+σZ.σAnimportantpropertyoftheNormalfamilyisthatalinearcombina-tionofindependentNormalvariablesresultsinaNormalvariable,thatis,ifX∼N(µ,σ2)andX∼N(µ,σ2)areindependentthenαX+βX∼11122212N(αµ+βµ,α2σ2+β2σ2).ThemomentgeneratingfunctionofXwith1212N(µ,σ2)distributionisgivenby∞22m(t)=EetX=etxf(x;µ,σ2)dx=eµte(σt)/2=eµt+(σt)/2.−∞ArandomvectorX=(X1,X2,...,Xn)hasann-variateNormal(Gaussian)distributionwithmeanvectorµandcovariancematrixΣifthereexistann×nmatrixAsuchthatitsdeterminant|A|=0,andX=µ+AZ,whereZ=(Z1,Z2,...,Zn)isthevectorwithindependentstandardNormalcomponents,andΣ=AAT.Vectorsaretakenascolumnvectorshereandinthesequel.TheprobabilitydensityofZisobtainedbyusingindependenceofitscom-ponents,forindependentrandomvariablesthedensitiesmultiply.Thenper-formingachangeofvariablesinthemultipleintegralwefindtheprobabilitydensityofX1−1(x−µ)Σ−1(x−µ)TfX(x)=e2.(2π)n/2|Σ|1/2"#1ρExample2.15:LetabivariateNormalhaveµ=0andΣ=.Letρ1X=(X,Y)andx=(x,y).ThenXcanbeobtainedfromZbythetransformation 42CHAPTER2.CONCEPTSOFPROBABILITYTHEORY"#10X=AZwithA=.ρ1−ρ2$x=z1Since,theinversetransformationy=ρz1+1−ρ2z2$z1=xhastheJacobianz2=(y−ρx)/1−ρ2"#"#∂z1∂z1101∂x∂yJ=det∂z2∂z2=det√1=.∂x∂y1−ρ21−ρ2ThedensityofZisgivenbytheproductofstandardNormaldensities,byinde-1−1(z2+z2)pendence,f(z1,z2)=e212.Usingtheformula(2.4)weobtainthejointZ2πdensityofthebivariateNormal1−1[x2−2ρxy+y2]fX(x,y)=e2(1−ρ2).2π1−ρ2ItfollowsfromthedefinitionthatifXhasamultivariateNormaldistributionandaisanon-randomvectorthenaX=a(µ+AZ)=aµ+aAZ.SincealinearcombinationofindependentNormalrandomvariablesisaNormalrandomvariable,aAZisaNormalrandomvariable.HenceaXhasNormaldistributionwithmeanaµandvariance(aA)(aA)T=aΣaT.ThuswehaveTheorem2.21AlinearcombinationofjointlyGaussianrandomvariablesisaGaussianrandomvariable.SimilarlyitcanbeshownthatifX∼N(µ,Σ)andBisanon-randommatrix,thenBX∼N(Bµ,BΣBT).ThemomentgeneratingfunctionofavectorXisdefinedasntXtiXiE(e)=E(ei=1),wheret=(t1,t2,...,tn),andtXisthescalarproductofvectorstandX.ItisnothardtoshowthatthemomentgeneratingfunctionofaGaussianvectorX∼N(µ,Σ)isgivenbyµtT−1tΣtTMX(t)=e2.Definition2.22AcollectionofrandomvariablesiscalledaGaussianpro-cess,ifthejointdistributionofanyfinitenumberofitsmembersisGaussian.Theorem2.23LetX(t)beaprocesswithindependentGaussianincrements,thatis,foranys0.However,suchanapproachfailsiftheeventweconditiononhaszeroprobability,P(Y=y)=0.ThisdifficultycanbeovercomeifX,Yhaveajointdensityf(x,y).Inthiscaseitfollowsthatboth∞XandYpossessdensitiesfX(x),andfY(y);fX(x)=−∞f(x,y)dy,and∞fY(y)=−∞f(x,y)dx.TheconditionaldistributionofXgivenY=yisdefinedbytheconditionaldensityf(x,y)f(x|y)=,fY(y)atanypointwherefY(y)>0.Itiseasytoseethatsodefinedf(x|y)isindeedaprobabilitydensityforanyy,asitisnon-negativeandintegratestounity.Theexpectationofthisdistribution,whenitexists,iscalledtheconditionalexpectationofXgivenY=y,∞E(X|Y=y)=xf(x|y)dx.(2.13)−∞TheconditionalexpectationE(X|Y=y)isafunctionofy.Letgdenotethisfunction,g(y)=E(X|Y=y),thenbyreplacingybyYweobtainarandomvariableg(Y),whichistheconditionalexpectationofXgivenY,E(X|Y)=g(Y).Example2.16:LetXandYhaveastandardbivariateNormaldistributionwithparameterρ.Then%&11221−y2/2f(x,y)=√exp−2(1−ρ2)[x−2ρxy+y],andfY(y)=√e,so2π1−ρ22πthat%&f(x,y)(x−ρy)2√12f(x|y)=fY(y)=2π(1−ρ2)exp−2(1−ρ2),whichistheN(ρy,1−ρ)distribu-tion.Itsmeanisρy,thereforeE(X|Y=y)=ρy,andE(X|Y)=ρY.Similarly,itcanbeseenthatinthemultivariateNormalcasetheconditionalexpectationisalsoalinearfunctionofY.TheconditionaldistributionandtheconditionalexpectationaredefinedonlyatthepointswherefY(y)>0.Bothcanbedefinedarbitrarilyontheset{y:fY(y)=0}.Sincetherearemanyfunctionswhichagreeontheset{y:fY(y)>0},anyoneofthemiscalledaversionoftheconditionaldistribution 44CHAPTER2.CONCEPTSOFPROBABILITYTHEORY(theconditionalexpectation)ofXgivenY=y.Thedifferentversionsoff(x|y)andE(X|Y=y)differonlyontheset{y:fY(y)=0},whichhaszeroprobabilityunderthedistributionofY;f(x|y)andE(X|Y=y)aredefineduniquelyY-almostsurely.GeneralConditionalExpectationTheconditionalexpectationinamoregeneralformisdefinedasfollows.LetXbeanintegrablerandomvariable.E(X|Y)=G(Y)afunctionofYsuchthatforanyboundedfunctionh,E(Xh(Y))=E(Yh(Y)),(2.14)orE(X−G(Y))h(Y)=0.ExistenceofsuchafunctionisassuredbytheRadon-Nikodymtheoremfromfunctionalanalysis.Butuniquenessiseasytoprove.Iftherearetwosuchfunctions,G1,G2,thenE((G1(Y)−G2(Y))h(Y))=0.Takeh(y)=sign(G1(y)−G2(y)).ThenwehaveE|G1(Y)−G2(Y)|=0.ThusP(G1(Y)=G2(Y))=1,andtheycoincidewith(Y)probabilityone.AmoregeneralconditionalexpectationofXgivenaσ-fieldG,E(X|G)isaG-measurablerandomvariablesuchthatforanyboundedG-measurableξE(ξE(X|G))=E(ξX).(2.15)Intheliterature,ξ=IBistakenasindicatorfunctionofasetB∈G,whichisanequivalentcondition:foranysetB∈GXdP=E(X|G)dP,orEXI(B)=EE(X|G)I(B).(2.16)BBTheRadon-Nikodymtheorem(seeTheorem10.6)impliesthatsucharan-domvariableexistsandisalmostsurelyunique,inthesensethatanytwoversionsdifferonlyonasetofprobabilityzero.TheconditionalexpectationE(X|Y)isgivenbyE(X|G)withG=σ(Y),theσ-fieldgeneratedbyY.OftentheEquations(2.15)or(2.16)arenotused,becauseeasiercalculationsarepossibletovariousspecificproperties,buttheyareusedtoestablishthefundamentalpropertiesgivenbelow.Inparticular,theconditionalexpectationdefinedin(2.13)byusingdensitiessatisfies(2.15)or(2.16).PropertiesofConditionalExpectationConditionalexpectationsarerandomvariables.Theirpropertiesarestatedasequalitiesoftworandomvariables.RandomvariablesXandY,definedonthesamespace,areequalifP(X=Y)=1.ThisisalsowrittenX=Y 2.7.CONDITIONALEXPECTATION45almostsurely(a.s.).Ifnotstatedotherwise,whenevertheequalityofrandomvariablesisuseditisunderstoodinthealmostsuresense,andoftenwriting“almostsurely”isomitted.1.IfGisthetrivialfield{∅,Ω},thenE(X|G)=EX,(2.17)2.IfXisG-measurable,thenE(XY|G)=XE(Y|G).(2.18)ThismeansthatifGcontainsalltheinformationaboutX,thengivenG,Xisknown,andthereforeitistreatedasaconstant.3.IfG1⊂G2thenE(E(X|G2)|G1)=E(X|G1).(2.19)Thisisknownasthesmoothingpropertyofconditionalexpectation.InparticularbytakingG1tobethetrivialfield,weobtainthelawofdoubleexpectationE(E(X|G))=E(X).(2.20)4.Ifσ(X)andGareindependent,thenE(X|G)=EX,(2.21)thatis,iftheinformationweknowprovidesnocluesaboutX,thentheconditionalexpectationisthesameastheexpectation.Thenextresultisanimportantgeneralization.5.Ifσ(X)andGareindependent,andFandGareindependent,andσ(F,G)denotesthesmallestσ-fieldcontainingbothofthem,thenE(X|σ(F,G))=E(X|F).(2.22)6.Jensen’sinequality.Ifg(x)isaconvexfunctiononI,thatis,forallx,y,∈Iandλ∈(0,1)g(λx+(1−λ)y)≤λg(x)+(1−λ)g(y),andXisarandomvariablewithrangeI,thengE(X|G)≤Eg(X)|G.(2.23)Inparticular,withg(x)=|x|'''E(X|G)'≤E|X||G.(2.24) 46CHAPTER2.CONCEPTSOFPROBABILITYTHEORY7.Monotoneconvergence.If0≤Xn,andXn↑XwithE|X|<∞,thenEXn|G↑EX|G.(2.25)8.Fatous’lemma.If0≤Xn,thenEliminfXn|G≤liminfEXn|G.(2.26)nn9.Dominatedconvergence.Iflimn→∞Xn=Xalmostsurelyand|Xn|≤YwithEY<∞,thenlimEXn|G=EX|G.(2.27)n→∞Forresultsonconditionalexpectationsseee.g.Breiman(1968),Chapter4.TheconditionalprobabilityP(A|G)isdefinedastheconditionalexpectationoftheindicatorfunction,P(A|G)=E(IA|G),anditisaG-measurablerandomvariable,definedP-almostsurely.Thefollowingresultsareoftenused.Theorem2.24LetXandYbetwoindependentrandomvariablesandφ(x,y)besuchthatE|φ(X,Y)|<+∞.ThenE(φ(X,Y)|Y)=G(Y),whereG(y)=E(φ(X,y)).Theorem2.25Let(X,Y)beaGaussianvector.Thentheconditionaldis-tributionofXgivenYisalsoGaussian.Moreover,providedthematrixCov(Y,Y)isnon-singular(hastheinverse),−1E(X|Y)=E(X)+Cov(X,Y)Cov(Y,Y)(Y−E(Y)).InthecasewhenCov(Y,Y)issingular,thesameformulaholdswiththeinversereplacedbythegeneralizedinverse,theMoore-Penrosepseudoinversematrix.Ifonewantstopredict/estimateXbyusingobservationsonY,thenapredictorissomefunctionofY.Forasquare-integrableX,E(X2)<∞,thebestpredictorXˆ,bydefinition,minimizesthemean-squareerror.ItiseasytoshowTheorem2.26(BestEstimator/Predictor)LetXˆbesuchthatforanyY-measurablerandomvariableZ,E(X−Xˆ)2≤E(X−Z)2.ThenXˆ=E(X|Y). 2.8.STOCHASTICPROCESSESINCONTINUOUSTIME472.8StochasticProcessesinContinuousTimeTheconstructionofamathematicalmodelofuncertaintyandofflowofin-formationincontinuoustimefollowsthesameideasasinthediscretetime,butitismuchmorecomplicated.ConsiderconstructingaprobabilitymodelforaprocessS(t)whentimechangescontinuouslybetween0andT.Takeforthesamplespacethesetofallpossibilitiesofmovementsoftheprocess.Ifwemakeasimplifyingassumptionthattheprocesschangescontinuously,weobtainthesetofallcontinuousfunctionson[0,T],denotedbyC[0,T].Thisisaveryrichspace.Inamoregeneralmodelitisassumedthattheobservedprocessisaright-continuousfunctionwithleftlimits(regularright-continuous(RRC,c`adl`ag))function.LetthesamplespaceΩ=D[0,T]bethesetofallRRCfunctionson[0,T].Anelementofthisset,ωisaRRCfunctionfrom[0,T]intoIR.Firstwemustdecidewhatkindofsetsofthesefunctionsaremeasurable.Thesimplestsetsforwhichwewouldliketocalculatetheprobabilitiesaresetsoftheform{a≤S(t1)≤b}forsomet1.IfS(t)representsthepriceofastockattimet,thentheprobabilityofsuchasetgivestheprobabilitythatthestockpriceattimet1isbetweenaandb.Wearealsointerestedinhowthepriceofstockattimet1affectsthepriceatanothertimet2.ThusweneedtotalkaboutthejointdistributionofstockpricesS(t1)andS(t2).Thismeansthatweneedtodefineprobabilityonthesetsoftheform{S(t1)∈B1,S(t2)∈B2}whereB1andB2areintervalsontheline.Moregenerallywewouldliketohaveallfinite-dimensionaldistributionsoftheprocessS(t),thatis,probabilitiesofthesets:{S(t1)∈B1,...,S(tn)∈Bn},foranychoiceof0≤t1≤t2,...≤tn≤T.Thesetsoftheform{ω(·)∈D[0,T]:ω(t1)∈B1,...,ω(tn)∈Bn},whereBi’sareintervalsontheline,arecalledcylindersetsorfinite-dimensionalrectangles.ThestochasticprocessS(t)onthissamplespaceisjusts(t),thevalueofthefunctionsatt.Probabilityisdefinedfirstonthecylindersets,andthenextendedtotheσ-fieldFgeneratedbythecylinders,thatis,thesmallestσ-fieldcontainingallcylindersets.Oneneedstobecarefulwithconsistencyofprobabilitydefinedoncylindersets,sothatwhenonecylindercontainsanothernocontradictionofprobabilityassignmentisobtained.Theresultthatshowsthataconsistentfamilyofdistributionsdefinesaprobabilityfunction,continuousat∅onthefieldofcylindersetsisknownasKolmogorov’sextensiontheorem.Onceaprobabilityisdefinedthefieldofcylindersets,itcanbeextendedinauniqueway(byCaratheodory’stheorem)toF(seeforexample,Breiman(1968),Durrett(1991)orDudley(1989)fordetails).Itfollowsimmediatelyfromthisconstructionthat:a)foranychoiceof0≤t1≤t2,...≤tn≤T,S(t1),S(t2),...,S(tn)isarandomvector,andb)thattheprocessisdeterminedbyitsfinite-dimensionaldistributions. 48CHAPTER2.CONCEPTSOFPROBABILITYTHEORYContinuityandRegularityofPathsAsdiscussedintheprevioussection,astochasticprocessisdeterminedbyitsfinite-dimensionaldistributions.Instudyingstochasticprocessesitisoftennaturaltothinkofthemasrandomfunctionsint.LetS(t)bedefinedfor0≤t≤T,thenforafixedωitisafunctionint,calledthesamplepathorarealizationofS.Finite-dimensionaldistributionsdonotdeterminethecontinuitypropertyofsamplepaths.Thefollowingexampleillustratesthis.Example2.17:LetX(t)=0forallt,0≤t≤1,andτbeauniformlydistributedrandomvariableon[0,1].LetY(t)=0fort=τandY(t)=1ift=τ.Thenforanyfixedt,P(Y(t)=0)=P(τ=t)=0,henceP(Y(t)=0)=1.Sothatallone-dimensionaldistributionsofX(t)andY(t)arethesame.Similarlyallfinite-dimensionaldistributionsofXandYarethesame.However,thesamplepathsoftheprocessX,thatis,thefunctionsX(t)0≤t≤1arecontinuousint,whereaseverysamplepathY(t)0≤t≤1hasajumpatthepointτ.NoticethatP(X(t)=Y(t))=1forallt,0≤t≤1.Definition2.27Twostochasticprocessesarecalledversions(modifications)ofoneanotherifP(X(t)=Y(t))=1forallt,0≤t≤T.ThusthetwoprocessesintheExample2.17areversionsofoneanother,onehascontinuoussamplepathsandtheotherdoesnot.Ifweagreetopickanyversionoftheprocesswewant,thenwecanpickthecontinuousversionwhenitexists.Ingeneralwechoosethesmoothestpossibleversionoftheprocess.FortwoprocessesXandY,denotebyNt={X(t)=Y(t)},0≤t≤T.IntheaboveExample2.17,P(Nt)=P(τ=t)=0foranyt,0≤t≤1.However,P(0≤t≤1Nt)=P(τ=tforsometin[0,1])=1.Although,eachofNtisaP-nullset,theunionN=0≤t≤1Ntcontainsuncountablymanynullsets,andinthiscaseitisasetofprobabilityone.IfithappensthatP(N)=0,thenNiscalledanevanescentset,andtheprocessesXandYarecalledindistinguishable.NotethatinthiscaseP({ω:∃t:X(t)=Y(t)})=P({X(t)=Y(t)})=0,and0≤t≤1P({X(t)=Y(t)})=P(X(t)=Y(t)forallt∈[0,T])=1.Itisclear0≤t≤1thatifthetimeisdiscretethenanytwoversionsoftheprocessareindistin-guishable.ItisalsonothardtoseethatifX(t)andY(t)areversionsofoneanotherandtheybothareright-continuous,thentheyareindistinguishable.Conditionsfortheexistenceofthecontinuousandtheregular(pathswithonlyjumpdiscontinuities)versionsofastochasticprocessaregivenbelow.Theorem2.28S(t),0≤t≤TisIR-valuedstochasticprocess.1.Ifthereexistα>0and >0,sothatforany0≤u≤t≤T,E|S(t)−S(u)|α≤C(t−u)1+,(2.28) 2.8.STOCHASTICPROCESSESINCONTINUOUSTIME49forsomeconstantC,thenthereexistsaversionofSwithcontinuoussamplepaths,whichareH¨oldercontinuousoforderh< /α.2.IfthereexistC>0,α1>0,α2>0and >0,sothatforany0≤u≤v≤t≤T,E|S(v)−S(u)|α1|S(t)−S(v)|α2≤C(t−u)1+,(2.29)thenthereexistsaversionofSwithpathsthatmayhavediscontinuitiesofthefirstkindonly(whichmeansthatatanyinteriorpointbothrightandleftlimitsexist,andone-sidedlimitsexistattheboundaries).Notethattheaboveresultallowstodecideontheexistenceofthecontinu-ous(regular)versionbymeansofthejointbivariate(trivariate)distributionsdoftheprocess.ThesameresultapplieswhentheprocesstakesvaluesinIR,exceptthattheEucledeandistancereplacestheabsolutevalueintheaboveconditions.Functionswithoutdiscontinuitiesofthesecondkindareconsideredtobethesameifatallpointsofthedomaintheyhavethesamerightandleftlimits.Inthiscaseitispossibletoidentifyanysuchfunctionwithitsright-continuousversion.Thefollowingresultgivesaconditionfortheexistenceofaregularright-continuousversionofastochasticprocess.Theorem2.29IfthestochasticprocessS(t)isright-continuousinprobability(thatis,foranytthelimitinprobabilitylimu↓tS(u)=S(t))anditdoesnothavediscontinuitiesofthesecondkind,thenithasaright-continuousversion.Otherconditionsfortheregularityofpathcanbegivenifweknowsomeparticularpropertiesoftheprocess.Forexample,laterwegivesuchconditionsforprocessesthataremartingalesandsupermartingales.σ-fieldGeneratedbyaStochasticProcessFt=σ(Su,u≤t)isthesmallestσ-fieldthatcontainssetsoftheform{a≤Su≤b}for0≤u≤t,a,b∈IR.ItistheinformationavailabletoanobserveroftheprocessSuptotimet.FilteredProbabilitySpaceandAdaptedProcessesAfiltrationIFisafamily{Ft}ofincreasingσ-fieldson(Ω,F),Ft⊂F.IFspecifieshowtheinformationisrevealedintime.Thepropertythatafiltrationisincreasingcorrespondstothefacttheinformationisnotforgotten. 50CHAPTER2.CONCEPTSOFPROBABILITYTHEORYIfwehaveasetΩ,aσ-fieldofsubsetsofΩ,F,aprobabilityPdefinedonelementsofF,andafiltrationIFsuchthatF0⊂Ft⊂...⊂FT=F,then(Ω,F,IF,P)iscalledafilteredprobabilityspace.Astochasticprocessonthisspace{S(t),0≤t≤T}iscalledadaptedifforallt,S(t)isFt-measurable,thatis,ifforanyt,FtcontainsalltheinformationaboutS(t)(andmaycontainextrainformation).TheUsualConditionsFiltrationiscalledright-continuousifFt+=Ft,whereFt+=Fs.s>tThestandardassumption(referredtoastheusualcondition)isthatfiltrationisright-continuous,forallt,Ft=Ft+.Ithasthefollowinginterpretation:anyinformationknownimmediatelyaftertisalsoknownatt.Remark2.2:NotethatifS(t)isaprocessadaptedtoIF,thenwecanalwaystakearight-continuousfiltrationtowhichS(t)isadaptedbytakingGt=Ft+=s>tFs.ThenStisGtadapted.Theassumptionofright-continuousfiltrationhasanumberofimportantconsequences.Forexample,itallowstoassumethatmartingales,submartin-galesandsupermartingaleshavearegularright-continuousversion.ItisalsoassumedthatanysetwhichisasubsetofasetofzeroprobabilityisF0-measurable.Ofcourse,suchasetmusthavezeroprobability.Apriorisuchsetsneednotbemeasurable,andweenlargetheσ-fieldstoincludesuchsets.Thisprocedureiscalledthecompletionbythenullsets.Martingales,Supermartingales,SubmartingalesDefinition2.30Astochasticprocess{X(t),t≥0}adaptedtoafiltrationIFisasupermartingale(submartingale)ifforanytitisintegrable,E|X(t)|<∞,andforanyst}∈Ft.Ifτt}={S(u)∈D,forallu≤t}=∩0≤u≤t{S(u)∈D}.Thiseventisanuncountableintersectionoverallu≤tofeventsinFu.Thepointoftheproofistorepresentthiseventasacountableintersection.DuetocontinuityofS(u)andDbeingopen,foranyirrationaluwithS(u)∈DthereisarationalqwithS(q)∈D.Therefore{S(u)∈D}={S(q)∈D},0≤u≤t0≤q−rational≤twhichisnowacountableintersectionoftheeventsfromFt,andhenceisitselfinFt.ThisshowsthatτDisastoppingtime.SinceforanyclosedsetA,IRAisopen,andTA=τIRA,TAisalsoastoppingtime. 2.8.STOCHASTICPROCESSESINCONTINUOUSTIME53Assumenowthatfiltrationisright-continuous.IfD=[a,b]isaclosedinterval,thenD=∩∞(a−1/n,b+1/n).IfD=(a−1/n,b+1/n),thenn=1nτDnisastoppingtime,andtheevent{τDn>t}∈Ft.Itiseasytoseethat∩∞{τ>t}={τ≥t},hence{τ≥t}∈F,andalso{τ0:S(t)∈A,orS(t−)∈A}isastoppingtime.Itispossible,althoughmuchharder,toshowthatthehittingtimeofaBorelsetisastoppingtime.Nextresultsgivebasicpropertiesofstoppingtimes.Theorem2.37LetSandTbetwostoppingtimes,thenmin(S,T),max(S,T),S+Tareallstoppingtimes.σ-fieldFTIfTisastoppingtime,eventsobservedbeforeorattimeTaredescribedbyσ-fieldFT,definedasthecollectionofsetsFT={A∈F:foranyt,A∩{T≤t}∈Ft}.Theorem2.38LetSandTbetwostoppingtimes.Thefollowingpropertieshold:IfA∈FS,thenA∩{S=T}∈FT,consequently{S=T}∈FS∩FT.IfA∈FS,thenA∩{S≤T}∈FT,consequently{S≤T}∈FS∩FT.Fubini’sTheoremFubini’stheoremallowsustointerchangeintegrals(sums)andexpectations.WegiveaparticularcaseofFubini’stheorem,itisformulatedinthewayweuseitinapplications.Theorem2.39LetX(t)beastochasticprocess0≤t≤T(foralltX(t)isarandomvariable),withregularsamplepaths(forallωatanypointt,X(t)hasleftandrightlimits).ThenTTE|X(t)|dt=E|X(t)|dt.00Furthermoreifthisquantityisfinite,thenTTEX(t)dt=E(X(t))dt.00 Thispageintentionallyleftblank Chapter3BasicStochasticProcessesThischapterismainlyaboutBrownianmotion.Itisthemainprocessinthecalculusofcontinuousprocesses.ThePoissonprocessisthemainprocessinthecalculusofprocesseswithjumps.Bothprocessesgiverisetofunctionsofpositivequadraticvariation.ForStochasticCalculusonlySection3.1-3.5areneeded,butinapplicationsothersectionsarealsoused.IntroductionObservationsofpricesofstocks,positionsofadiffusingparticleandmanyotherprocessesobservedintimeareoftenmodelledbyastochasticprocess.Astochasticprocessisanumbrellatermforanycollectionofrandomvariables{X(t)}dependingontimet.Timecanbediscrete,forexample,t=0,1,2,...,orcontinuous,t≥0.Calculusissuitedmoretocontinuoustimeprocesses.Atanytimet,theobservationisdescribedbyarandomvariablewhichwedenotebyXtorX(t).Astochasticprocess{X(t)}isfrequentlydenotedbyXorwithaslightabuseofnotationalsobyX(t).Inpractice,wetypicallyobserveonlyasinglerealizationofthisprocess,asinglepath,outofamultitudeofpossiblepaths.Anysinglepathisafunctionoftimet,xt=x(t),0≤t≤T;andtheprocesscanalsobeseenasarandomfunction.Todescribethedistributionandtobeabletodoprobabilitycalculationsabouttheuncertainfuture,oneneedstoknowtheso-calledfinite-dimensionaldistributions.Namely,weneedtospecifyhowtocalculateprobabilitiesoftheformP(X(t)≤x)foranytimet,i.e.theprobabilitydistributionoftherandomvariableX(t);andprobabilitiesoftheformP(X(t1)≤x1,X(t2)≤x2)foranytimest1,t2,i.e.thejointbivariatedistributionsofX(t1)andX(t2);andprobabilitiesoftheformP(X(t1)≤x1,X(t2)≤x2,...X(tn)≤xn),(3.1)55 56CHAPTER3.BASICSTOCHASTICPROCESSESforanychoiceoftimepoints0≤t1s,isindependentofthepast,thatis,ofBu,0≤u≤s,orofFs,theσ-fieldgeneratedbyB(u),u≤s.2.(Normalincrements)B(t)−B(s)hasNormaldistributionwithmean0andvariancet−s.Thisimplies(takings=0)thatB(t)−B(0)hasN(0,t)distribution.3.(Continuityofpaths)B(t),t≥0arecontinuousfunctionsoft.TheinitialpositionofBrownianmotionisnotspecifiedinthedefinition.WhenB(0)=x,thentheprocessisBrownianmotionstartedatx.Properties1and2abovedetermineallthefinite-dimensionaldistributions(see(3.4)be-low)anditispossibletoshow(seeTheorem3.3)thatallofthemareGaussian.Pxdenotestheprobabilityofeventswhentheprocessstartsatx.ThetimeintervalonwhichBrownianmotionisdefinedis[0,T]forsomeT>0,whichisallowedtobeinfinite.Wedon’tproveherethataBrownianmotionexists,itcanbefoundinmanybooksonstochasticprocesses,andoneconstructionisoutlinedinSection5.7. 3.1.BROWNIANMOTION57HoweverwecandeducecontinuityofpathsbyusingnormalityofincrementsandappealingtoTheorem2.28.SinceE(B(t)−B(s))4=3(t−s)2,acontinuousversionofBrownianmotionexists.Remark3.1:AdefinitionofBrownianmotioninamoregeneralmodel(thatcontainsextrainformation)isgivenbyapair{B(t),Ft},t≥0,whereFtisanincreasingsequenceofσ-fields(afiltration),B(t)isanadaptedprocess,i.e.B(t)isFtmeasurable,suchthatProperties1-3abovehold.Animportantrepresentationusedforcalculationsinprocesseswithinde-pendentincrementsisthatforanys≥0B(t+s)=B(s)+(B(t+s)−B(s)),(3.2)wheretwovariablesareindependent.Anextensionofthisrepresentationistheprocessversion.LetW(t)=B(t+s)−B(s).Thenforafixeds,asaprocessint,W(t)isaBrownianmotionstartedat0.Thisisseenbyverifyingthedefiningproperties.OtherexamplesofBrownianmotionprocessesconstructedfromotherpro-cessesaregivenbelow,aswellasinexercises.Example3.1:AlthoughB(t)−B(s)isindependentofthepast,2B(t)−B(s)orB(t)−2B(s)isnot,as,forexample,B(t)−2B(s)=(B(t)−B(s))−B(s),isasumoftwovariables,withonlyoneindependentofthepastandB(s).ThefollowingexampleillustratescalculationsofsomeprobabilitiesforBrow-nianmotion.Example3.2:LetB(0)=0.WecalculateP(B(t)≤0fort=2)andP(B(t)≤0fort=0,1,2).SinceB(2)hasNormaldistributionwithmeanzeroandvariance2,1P(B(t)≤0fort=2)=.2SinceB(0)=0,P(B(t)≤0fort=0,1,2)=P(B(1)≤0,B(2)≤0).NotethatB(2)andB(1)arenotindependent,thereforethisprobabilitycannotbecalculatedasaproductP(B(1)≤0)P(B(2)≤0)=1/4.UsingthedecompositionB(2)=B(1)+B(2)−B(1)=B(1)+W(1),wherethetworandomvariablesareindependent,wehaveP(B(1)≤0,B(2)≤0)=P(B(1)≤0,B(1)+W(1)≤0)=P(B(1)≤0,W(1)≤−B(1)).ByconditioningandbyusingTheorem2.24and(2.20)00P(B(1)≤0,W(1)≤−B(1))=P(W(1)≤−x)f(x)dx=Φ(−x)dΦ(x),−∞−∞ 58CHAPTER3.BASICSTOCHASTICPROCESSESwhereΦ(x)andf(x)denotethedistributionandthedensityfunctionsofthestandardNormaldistribution.Bychangingvariablesthelastintegral,weobtain∞∞13Φ(x)f(−x)dx=Φ(x)dΦ(x)=ydy=.8001/2TransitionProbabilityFunctionsIftheprocessisstartedatx,B(0)=x,thenB(t)hastheN(x,t)distribution.Moregenerally,theconditionaldistributionofB(t+s)giventhatB(s)=xisN(x,t).ThetransitionfunctionP(y,t,x,s)isthecumulativedistributionfunctionofthisdistribution,P(y,t,x,s)=P(B(t+s)≤y|B(s)=x)=Px(B(t)≤y).Thedensityfunctionofthisdistributionisthetransitionprobabilityden-sityfunctionofBrownianmotion,21−(y−x)pt(x,y)=√e2t.(3.3)2πtThefinite-dimensionaldistributionscanbecomputedwiththehelpofthetransitionprobabilitydensityfunction,byusingindependenceofincrementsinawaysimilartothatexhibitedintheaboveexample.Px(B(t1)≤x1,B(t2)≤x2,...,B(tn)≤xn)=(3.4)x1x2xnpt1(x,y1)dy1pt2−t1(y1,y2)dy2...ptn−tn−1(yn−1,yn)dyn.−∞−∞−∞SpaceHomogeneityItiseasytoseethattheone-dimensionaldistributionsofBrownianmotionsatisfyP0(B(t)∈A)=Px(B(t)∈x+A),whereAisanintervalontheline.IfBx(t)denotesBrownianmotionstartedatx,thenitfollowsfrom(3.4)thatallfinite-dimensionaldistributionsofBx(t)andx+B0(t)arethesame.ThusBx(t)−xisBrownianmotionstartedat0,andB0(t)+xisBrownianmotionstartedatx,inotherwordsBx(t)=x+B0(t).(3.5)Theproperty(3.5)iscalledthespacehomogeneouspropertyofBrownianmo-tion.Definition3.1Astochasticprocessiscalledspace-homogeneousifitsfinite-dimensionaldistributionsdonotchangewithashiftinspace,namelyifP(X(t1)≤x1,X(t2)≤x2,...X(tn)≤xn|X(0)=0)=P(X(t1)≤x1+x,X(t2)≤x2+x,...X(tn)≤xn+x|X(0)=x). 3.1.BROWNIANMOTION59FourrealizationsofBrownianmotionB=B(t)startedat0areexhibitedinFigure3.1.Althoughitisaprocessgovernedbythepurechancewithzeromean,ithasregionswheremotionlookslikeithas“trends”.-1.00.00.00.51.00.00.20.40.60.81.00.00.20.40.60.81.0timetime0.00.40.8-0.40.00.40.00.20.40.60.81.00.00.20.40.60.81.0timetimeFigure3.1:FourrealizationsorpathsofBrownianmotionB(t).BrownianMotionasaGaussianProcessRecallthataprocessiscalledGaussianifallitsfinite-dimensionaldistributionsaremultivariateNormal.Example3.3:LetrandomvariablesXandYbeindependentNormalwithdistributionsN(µ,σ2)andN(µ,σ2).Thenthedistributionof(X,X+Y)1122"isbivariateNormalwithmeanvector(#µ1,µ1+µ2)andcovariancematrixσ2σ211.σ2σ2+σ2112ToseethisletZ=(Z1,Z2)havestandardNormalcomponents,thenitiseasytoseethat(X,X+Y)=µ+AZ,"#σ10whereµ=(µ1,µ1+µ2),andmatrixA=.Theresultfollowsbyσ1σ2thedefinitionofthegeneralNormaldistributionasalineartransformationofstandardNormals(seeSection2.6). 60CHAPTER3.BASICSTOCHASTICPROCESSESSimilarlytotheaboveexample,thefollowingrepresentation(B(t1),B(t2),...,B(tn))=(B(t1),B(t1)+(B(t2)−B(t1)),...,B(tn−1))+(B(tn))−B(tn−1))showsthatthisvectorisalineartransformationofthestandardNormalvector,henceithasamultivariateNormaldistribution.LetY1=B(t1),andfork>1,Yk=B(tk)−B(tk−1).Thenbytheprop-ertyofindependenceofincrementsofBrownianmotion,Yk’sareindependent.TheyalsohaveNormaldistribution,Y1∼N(0,t1),andYk∼N(0,tk−tk−1).Thus(B(t√1),B(t2),...,B(tn√))isalineartransformationof(Y1,Y2,...,Yn).ButY1=t1Z1,andYk=tk−tk−1Zk,whereZk’sareindependentstan-dardNormal.Thus(B(t1),B(t2),...,B(tn))isalineartransformationof(Z1,...,Zn).FindingthematrixAofthistransformationisleftasanex-ercise(Exercise3.7).Definition3.2ThecovariancefunctionoftheprocessX(t)isdefinedbyγ(s,t)=CovX(t),X(s)=EX(t)−EX(t)X(s)−EX(s)=EX(t)X(s)−EX(t)EX(s).(3.6)ThenextresultcharacterizesBrownianmotionasaparticularGaussianprocess.Theorem3.3ABrownianmotionisaGaussianprocesswithzeromeanfunc-tion,andcovariancefunctionmin(t,s).Conversely,aGaussianprocesswithzeromeanfunction,andcovariancefunctionmin(t,s)isaBrownianmotion.Proof:SincethemeanoftheBrownianmotioniszero,γ(s,t)=CovB(t),B(s)=EB(t)B(s).Ifts,E(B(t)B(s))=s.ThereforeE(B(t)B(s))=min(t,s).Toshowtheconverse,lettbearbitraryands≥0.X(t)isaGaussianprocess,thusthejointdistributionofX(t),X(t+s)isabivariateNormal,andbyconditionshaszeromean.Thereforethevector(X(t),X(t+s)−X(t)isalso 3.1.BROWNIANMOTION61bivariateNormal.ThevariablesX(t)andX(t+s)−X(t)areuncorrelated,usingthatCov(X(t),X(t+s))=min(t,s),Cov(X(t),X(t+s)−X(t))=Cov(X(t),X(t+s))−Cov(X(t),X(t))=t−t=0.ApropertyofthemultivariateNormaldistributionimpliesthatthesevariablesareindependent.ThustheincrementX(t+s)−X(t)isindependentofX(t)andhasN(0,s)distribution.ThereforeitisaBrownianmotion.
Example3.4:WefindthedistributionofB(1)+B(2)+B(3)+B(4).ConsidertherandomvectorX=(B(1),B(2),B(3),B(4)).SinceBrownianmotionisaGaussianprocess,allitsfinite-dimensionaldistributionsareNormal,inparticularXhasamultivariateNormaldistributionwithmeanvectorzeroandcovariancematrixgivenbyσij=Cov(Xi,Xj).Forexample,Cov(X1,X3)=Cov((B(1),B(3))=1.11111222Σ=12331234Now,leta=(1,1,1,1).ThenaX=X1+X2+X3+X4=B(1)+B(2)+B(3)+B(4).TaXhasaNormaldistributionwithmeanzeroandvarianceaΣa,andinthiscasethevarianceisgivenbythesumoftheelementsofthecovariancematrix.ThusB(1)+B(2)+B(3)+B(4)hasaNormaldistributionwithmeanzeroandvariance30.Alternatively,wecancalculatethevarianceofthesumbytheformulaVar(X1+X2+X3+X4)
=Cov(X1+X2+X3+X4,X1+X2+X3+X4)=Cov(Xi,Xj)=30.i,jExample3.5:Toillustratetheuseofscaling,wewefindthedistributionof113113B()+B()+B()+B(1).ConsidertherandomvectorY=B(),B(),B(),B(1).424424ItiseasytoseethatYand1/2X,whereX=(B(1),B(2),B(3),B(4))havethesame1law.ThereforeitscovariancematrixisgivenbyΣ,withΣasabove.Consequently,4aYhasaNormaldistributionwithmeanzeroandvariance30/412Example3.6:WefindtheprobabilityP(B(t)dt>√).03CommentfirstthatsinceBrownianmotionhascontinuouspaths,theRiemannin-1tegralB(t)dtiswelldefinedforanyrandompathasweintegratepathbypath.01TofindtherequiredprobabilityweneedtoknowthedistributionofB(t)dt.This0canbeobtainasalimitofthedistributionsoftheapproximatingsums,
B(ti)∆, 62CHAPTER3.BASICSTOCHASTICPROCESSESwherepointstipartition[0,1]and∆=ti+1−ti.If,forexample,ti=i/n,thenfor1113n=4theapproximatingsumisB()+B()+B()+B(1),thedistributionof442415whichwasfoundinthepreviousexampletobeN(0,).Similarly,thedistribution32ofalloftheapproximatingsumsisNormalwithzeromean.Itcanbeshownthat1thelimitofGaussiandistributionsisaGaussiandistribution.ThusB(t)dthas0aNormaldistributionwithzeromean.Thereforeitonlyremainstocomputeitsvariance.)*)*111VarB(t)dt=CovB(t)dt,B(s)ds000)*1111=EB(t)dtB(s)ds=E(B(t)B(s))dtds00001111=Cov(B(t),B(s))dtds=min(t,s)dtds=1/30000ExchangingtheintegralsandexpectationisjustifiedbyFubini’stheoremsince11''11√E'B(t)B(s)'dtds≤tsdtds<1.00001ThusB(t)dthasN(0,1/3)distribution,andthedesiredprobabilityisapproxi-0amately0.025.LaterweshallprovethatthedistributionoftheintegralB(t)dtis03NormalN(0,a/3)byconsideringatransformationtoItˆointegral,seeExample6.4.BrownianMotionasaRandomSeriesTheprocess
∞t2sin(jt)ξ0√+√ξj,(3.7)ππjj=1whereξj’sj=0,1,...,areindependentstandardNormalrandomvariables,isBrownianmotionon[0,π].Convergenceoftheseriesisunderstoodalmostsurely.Thisrepresentationresemblestheexampleofacontinuousbutnowheredifferentiablefunction,Example1.2.Onecanprovetheassertionbyshowingthatthepartialsumsconvergeuniformly,andverifyingthattheprocessin(3.7)isGaussian,haszeromean,andcovariancemin(s,t)(see,forexample,Breiman(1968),p.261,ItˆoandMcKean(1965),p.22).Remark3.2:Asimilar,moregeneralrepresentationofaBrownianmotionisgivenbyusinganorthonormalsequenceoffunctionson[0,T],hj(t).B(t)=∞tj=0ξjHj(t),whereHj(t)=0hj(s)ds,isaBrownianmotionon[0,T]. 3.2.PROPERTIESOFBROWNIANMOTIONPATHS633.2PropertiesofBrownianMotionPathsAnoccurrenceofBrownianmotionobservedfromtime0totimeT,isarandomfunctionoftontheinterval[0,T].Itiscalledarealization,apathortrajectory.QuadraticVariationofBrownianMotionThequadraticvariationofBrownianmotion[B,B](t)isdefinedas
nnn2[B,B](t)=[B,B]([0,t])=lim|B(ti)−B(ti−1)|,(3.8)i=1wherethelimitistakenoverallshrinkingpartitionsof[0,t],withδn=max(tn−tn)→0asn→∞.Itisremarkablethatalthoughthesumsii+1iinthedefinition(3.8)arerandom,theirlimitisnon-random,asthefollowingresultshows.Theorem3.4QuadraticvariationofaBrownianmotionover[0,t]ist.Proof:Wegivetheproofforasequenceofpartitions,forwhichnδn<∞.Anexampleofsuchiswhentheintervalisdividedintotwo,theneachsubintervalisdividedintotwo,etc.LetT=|B(tn)−B(tn)|2.Itiseasyniii−1toseethat

nE(T)=E|B(tn)−B(tn)|2=(tn−tn)=t−0=t.nii−1ii−1ii=1ByusingthefourthmomentofN(0,σ2)distributionis3σ4,weobtainthevarianceofTn

nn2nn2Var(Tn)=Var(|B(ti)−B(ti−1)|)=Var(B(ti)−B(ti−1))ii
=3(tn−tn)2≤3max(tn−tn)t=3tδ.ii−1ii−1ni∞Thereforen=1Var(Tn)<∞.Usingmonotoneconvergencetheorem,wefindE∞(T−ET)2<∞.Thisimpliesthattheseriesinsidetheexpectationn=1nnconvergesalmostsurely.Henceitstermsconvergetozero,andTn−ETn→0a.s.,consequentlyTn→ta.s.ItispossibletoshowthatTn→ta.s.foranysequenceofpartitionswhicharesuccessiverefinementsandsatisfyδn→0asn→∞(seeforexample,Loeve(1978),Vol.2,p.253fortheproof,orBreiman(1968)).
64CHAPTER3.BASICSTOCHASTICPROCESSESVaryingt,thequadraticvariationprocessofBrownianmotionist.NotethattheclassicalquadraticvariationofBrownianpaths(definedasthesupremumoverallpartitionsofsumsin(3.8),seeChapter1isinfinite(e.g.Freedman(1971),p.48.)PropertiesofBrownianpathsB(t)’sasfunctionsofthavethefollowingproperties.AlmosteverysamplepathB(t),0≤t≤T1.isacontinuousfunctionoft;2.isnotmonotoneinanyinterval,nomatterhowsmalltheintervalis;3.isnotdifferentiableatanypoint;4.hasinfinitevariationonanyinterval,nomatterhowsmallitis;5.hasquadraticvariationon[0,t]equaltot,foranyt.Properties1and3ofBrownianmotionpathsstatethatalthoughanyrealiza-tionB(t)isacontinuousfunctionoft,ithasincrements∆B(t)overanintervaloflength∆tmuchlargerthan∆tas∆t→0.SinceE(B(t+∆t)−B(t))2=∆t,√itsuggeststhattheincrementisroughlylike∆t.ThisismadeprecisebythequadraticvariationProperty5.NotethatbyTheorem1.10,apositivequadraticvariationimpliesinfinitevariation,sothatProperty4followsfromProperty5.Sinceamonotonefunctionhasfinitevariation,Property2followsfromProperty4.ByTheorem1.8acontinuousfunctionwithaboundedderivativeisoffinitevariation.ThereforeitfollowsfromProperty4thatB(t)cannothaveaboundedderivativeonanyinterval,nomatterhowsmalltheintervalis.Itisnotyetthenon-differentiabilityatanypoint,butitisclosetoit.FortheproofoftheresultthatwithprobabilityoneBrownianmotionpathsarenowheredifferentiable(duetoDvoretski,Erd¨osandKakutani)seeBreiman(1968)p.261.HereweshowasimplestatementTheorem3.5ForanytalmostalltrajectoriesofBrownianmotionarenotdifferentiableatt.√Proof:ConsiderB(t+∆)−B(t)=∆Z=√Z,forsomestandardNormal∆∆∆randomvariableZ.Thustheratioconvergesto∞indistribution,sinceP(|√Z|>K)→1foranyK,as∆→0,precludingexistenceofthe∆derivativeatt.
Torealizetheaboveargumentonacomputertakee.g.∆=10−20.Then∆B(t)=10−10Z,and∆B(t)/∆=1010Z,whichisverylargeinabsolutevaluewithoverwhelmingprobability. 3.3.THREEMARTINGALESOFBROWNIANMOTION653.3ThreeMartingalesofBrownianMotionInthissectionthreemainmartingalesassociatedwithBrownianmotionaregiven.Recallthedefinitionofamartingale.Definition3.6Astochasticprocess{X(t),t≥0}isamartingaleifforanytitisintegrable,E|X(t)|<∞,andforanys>0E(X(t+s)|Ft)=X(t),a.s.(3.9)whereFtistheinformationabouttheprocessuptotimet,andtheequalityholdsalmostsurely.Themartingalepropertymeansthatifweknowthevaluesoftheprocessuptotimet,andX(t)=xthentheexpectedfuturevalueatanyfuturetimeisx.Remark3.3:Ftrepresentsinformationavailabletoanobserverattimet.AsetA∈FtifandonlyifbyobservingtheprocessuptotimetonecandecidewhetherornotAhasoccurred.Formally,Ft=σ(X(s),0≤s≤t)denotestheσ-field(σ-algebra)generatedbythevaluesoftheprocessuptotimet.Remark3.4:Astheconditionalexpectationgivenaσ-fieldisdefinedasarandomvariable(seeforexample,Section2.7),alltherelationsinvolvingcon-ditionalexpectations,suchasequalitiesandinequalities,mustbeunderstoodinthealmostsuresense.Thiswillalwaysbeassumed,andthealmostsure“a.s.”specificationwillbefrequentlydropped.ExamplesofmartingalesconstructedfromBrownianmotionaregiveninthenextresult.Theorem3.7LetB(t)beBrownianMotion.Then1.B(t)isamartingale.2.B(t)2−tisamartingale.u2uB(t)−t3.Foranyu,e2isamartingale.Proof:Thekeyideainestablishingthemartingalepropertyisthatforanyfunctiong,theconditionalexpectationofg(B(t+s)−B(t))givenFtequalstotheunconditionalone,Eg(B(t+s)−B(t))|Ft=Eg(B(t+s)−B(t)),(3.10) 66CHAPTER3.BASICSTOCHASTICPROCESSESduetoindependenceofB(t+s)−B(t)andFt.ThelatterexpectationisjustEg(X),whereXNormalN(0,s)randomvariable.1.Bydefinition,B(t)∼N(0,t),sothatB(t)isintegrablewithE(B(t))=0.E(B(t+s)|Ft)=EB(t)+(B(t+s)−B(t))|Ft=E(B(t)|Ft)+E(B(t+s)−B(t)|Ft)=B(t)+E(B(t+s)−B(t))=B(t).2.Bydefinition,E(B2(t))=t<∞,thereforeB2(t)isintegrable.Since2B2(t+s)=B(t)+B(t+s)−B(t)=B2(t)+2B(t)(B(t+s)−B(t))+(B(t+s)−B(t))2,E(B2(t+s)|F)t=B2(t)+2EB(t)(B(t+s)−B(t))|F+E(B(t+s)−B(t))2|Ftt=B2(t)+s,whereweusedthatB(t+s)−B(t)isindependentofFtandhasmean0,and(3.10)withg(x)=x2.Subtracting(t+s)frombothsidesgivesthemartingalepropertyofB2(t)−t.3.ConsiderthemomentgeneratingfunctionofB(t),2E(euB(t))=etu/2<∞,2sinceB(t)hastheN(0,t)distribution.ThisimpliesintegrablityofeuB(t)−tu/2,moreover2E(euB(t)−tu/2)=1.Themartingalepropertyisestablishedbyusing(3.10)withg(x)=eux.EeuB(t+s)|F=EeuB(t)+u(B(t+s)−B(t))|Ftt=euB(t)Eeu(B(t+s)−B(t))|F(sinceB(t)isF-measurable)ttuB(t)u(B(t+s)−B(t))=eEe(sinceincrementisindependentofFt)u2suB(t)=e2e.2ThemartingalepropertyofeuB(t)−tu/2isobtainedbymultiplyingbothsides2−u(t+s)bye2.
3.4.MARKOVPROPERTYOFBROWNIANMOTION67Remark3.5:Allthreemartingaleshaveacentralplaceinthetheory.ThemartingaleB2(t)−tprovidesacharacterization(Levy’scharacterization)ofBrownianmotion.ItwillbeseenlaterthatifaprocessX(t)isacontinuousmartingalesuchthatX2(t)−tisalsoamartingale,thenX(t)isBrownian2motion.ThemartingaleeuB(t)−tu/2isknownastheexponentialmartingale,andasitisrelatedtothemomentgeneratingfunction,itisusedforestablishingdistributionalpropertiesoftheprocess.3.4MarkovPropertyofBrownianMotionTheMarkovPropertystatesthatifweknowthepresentstateoftheprocess,thenthefuturebehaviouroftheprocessisindependentofitspast.TheprocessX(t)hastheMarkovpropertyiftheconditionaldistributionofX(t+s)givenX(t)=x,doesnotdependonthepastvalues(butitmaydependonthepresentvaluex).Theprocess“doesnotremember”howitgottothepresentstatex.LetFtdenotetheσ-fieldgeneratedbytheprocessuptotimet.Definition3.8XisaMarkovprocessifforanytands>0,theconditionaldistributionofX(t+s)givenFtisthesameastheconditionaldistributionofX(t+s)givenX(t),thatis,P(X(t+s)≤y|Ft)=P(X(t+s)≤y|X(t)),a.s.(3.11)Theorem3.9BrownianmotionB(t)possessesMarkovproperty.Proof:ItiseasytoseebyusingthemomentgeneratingfunctionthattheconditionaldistributionofB(t+s)givenFtisthesameasthatgivenB(t).Indeed,E(euB(t+s)|F)=euB(t)Eeu(B(t+s)−B(t))|FttuB(t)u(B(t+s)−B(t))u(B(t+s)−B(t))=eEe(sinceeisindependentofFt)2=euB(t)eus/2(sinceB(t+s)−B(t)isN(0,s))=euB(t)Eeu(B(t+s)−B(t))|B(t)=EeuB(t+s)|B(t).
ThetransitionprobabilityfunctionofaMarkovprocessXisdefinedasP(y,t,x,s)=P(X(t)≤y|X(s)=x)theconditionaldistributionfunctionoftheprocessattimet,giventhatitisatpointxattimes0:B(t)=a}isastoppingtime.4.LetTbethetimewhenBrownianmotionreachesitsmaximumontheinterval[0,1].Thenclearly,todecidewhether{T≤t}hasoccurredornot,itisnotenoughtoknowthevaluesoftheprocesspriortot,oneneedstoknowallthevaluesontheinterval[0,1].SothatTisnotastoppingtime.5.LetTbethelastzeroofBrownianmotionbeforetimet=1.ThenTisnotastoppingtime,sinceifT≤t,thentherearenozerosin(t,1],whichistheeventthatisdecidedbyobservingtheprocessuptotime1,andthissetdoesnotbelongtoFt. 3.5.HITTINGTIMESANDEXITTIMES69ThestrongMarkovpropertyissimilartotheMarkovproperty,exceptthatinthedefinitionafixedtimetisreplacedbyastoppingtimeT.Theorem3.11BrownianmotionB(t)hastheStrongMarkovproperty:foranyfinitestoppingtimeTtheregularconditionaldistributionofB(T+t),t≥0givenFTisPB(T),thatis,P(B(T+t)≤y|FT)=P(B(T+t)≤y|B(T))a.s.Corollary3.12LetTbeafinitestoppingtime.Definethenewprocessint≥0byBˆ(t)=B(T+t)−B(T).(3.13)ThenBˆ(t)isaBrownianmotionisstartedatzeroandindependentofFT.Wedon’tgivetheproofofthestrongMarkovpropertyhere,itcanbefound,forexampleinRogersandWilliams(1994)p.21,andcanbedonebyusingtheexponentialmartingaleandtheOptionalStoppingTheoremgiveninChapter7.NotethatthestrongMarkovpropertyappliesonlywhenTisastoppingtime.IfTisjustarandomtime,thenB(T+t)−B(T)neednotbeBrownianmotion.3.5HittingTimesandExitTimesLetTxdenotethefirsttimeB(t)hitslevelx,Tx=inf{t>0:B(t)=x}.Denotethetimetoexitaninterval(a,b)byτ=min(Ta,Tb).Theorem3.13Leta1}={a1)≤Px(B(1)∈(a,b))=√edy.2πaThefunctionPx(B(1)∈(a,b))iscontinuousinxon[a,b],henceitreachesitsmaximumθ<1.ByusingthestrongMarkovpropertywecanshowthatP(τ>n)≤θn.Foranynon-negativerandomvariableX≥0,EX≤x∞P(X>n)(seeExercise3.2).Therefore,n=0
∞n1Exτ≤θ=<∞.1−θn=0 70CHAPTER3.BASICSTOCHASTICPROCESSESTheboundonPx(τ>n)isestablishedasfollowsPx(τ>n)=Px(B(s)∈(a,b),0≤s≤n)=Px(B(s)∈(a,b),0≤s≤n−1,B(s)∈(a,b),n−1≤s≤n)=Px(τ>n−1,B(s)∈(a,b),n−1≤s≤n)=Px(τ>n−1,B(n−1)+Bˆ(s)∈(a,b),0≤s≤1)by(3.13)y=E(Px(τ>n−1,Bˆ(s)∈(a,b),0≤s≤1|B(n−1)=y))y=E((Px(τ>n−1|B(n−1)=y)Px(Bˆ(s)∈(a,b),0≤s≤1)|B(n−1)=y))=E((Px(τ>n−1|B(n−1)=y))Py(Bˆ(s)∈(a,b),0≤s≤1))≤maxPy(Bˆ(s)∈(a,b),0≤s≤1)Px(τ>n−1)n≤θPx(τ>n−1)≤θ,byiterations.
ThenextresultgivestherecurrencepropertyofBrownianmotion.Theorem3.14Pa(Tb<∞)=1,Pa(Ta<∞)=1Proof:Thesecondstatementfollowsfromthefirst,sincePa(Ta<∞)≥Pa(Tb<∞)Pb(Ta<∞)=1.WeshownowthatP0(T1<∞)=1,forotherpointstheproofissimilar.Observefirstlythatbythepreviousresultandbysymmetry,foranyaandb1Pa+b(Ta0:B(t)=x}.Theorem3.15Foranyx>0,xP0(M(t)≥x)=2P0(B(t)≥x)=2(1−Φ(√)),twhereΦ(x)standsforthestandardNormaldistributionfunction.Proof:Noticethattheevents{M(t)≥x}and{Tx≤t}arethesame.Indeed,ifthemaximumattimetisgreaterthanx,thenatsometimebeforetBrownianmotiontookvaluex,andifBrownianmotiontookvaluexatsometimebeforet,thenthemaximumwillbeatleastx.Since{B(t)≥x}⊂{Tx≤t}P(B(t)≥x)=P(B(t)≥x,Tx≤t).AsB(Tx)=x,P(B(t)≥x)=P(Tx≤t,B(Tx+(t−Tx))−B(Tx)≥0). 72CHAPTER3.BASICSTOCHASTICPROCESSESByTheorem3.14,Txisafinitestoppingtime,andbythestrongMarkovproperty(3.13),therandomvariableBˆ(s)=B(Tx+s)−B(Tx)isindependentofFTxandhasaNormaldistribution,sowehaveP(B(t)≥x)=P(Tx≤t,Bˆ(t−Tx)≥0).(3.14)IfwehadsindependentofTx,thenP(Tx≤t,Bˆ(s)≥0)=P(Tx≤t)P(Bˆ(s≥0)11=P(Tx≤t)=P(M(t)≥x),(3.15)22andwearedone.Butin(3.14)s=t−Tx,andisclearlydependentonTx.ItisnoteasytoshowthatP(B(t)≥x)=P(Tx≤t,Bˆ(t−Tx)≥0)11=P(Tx≤t)=P(M(t)≥x).22Theproofcanbefound,forexample,inDudley(1989)p.361.
Asimpleapplicationoftheresultisgiveninthefollowingexample,fromwhichitfollowsthatBrownianmotionchangessignin(0,ε),foranyεhoweversmall.Example3.8:WefindtheprobabilityP(B(t)≤0forallt,0≤t≤1).Notethattherequiredprobabilityinvolvesuncountablymanyrandomvariables:allB(t)’sarelessthanorequaltozero,0≤t≤1,wewanttoknowtheprobabilitythattheentirepathfrom0to1willstaybelow0.Wecouldcalculatethedesiredprobabilityfornvaluesoftheprocessandthentakethelimitasn→∞.Butitissimplerinthiscasetoexpressthisprobabilityasafunctionofthewholepath.AllB(t)’sarelessorequalzero,ifandonlyiftheirmaximumislessthanorequaltozero.{B(t)≤0forallt,0≤t≤1}={maxB(t)≤0},0≤t≤1andconsequentlytheseeventshavesameprobabilities.Now,P(maxB(t)≤0)=1−P(maxB(t)>0).0≤t≤10≤t≤1BythelawofthemaximumofBrownianmotion,P(maxB(t)>0)=2P(B(1)>0)=1.0≤t≤1HenceP(B(t)≤0forallt,0≤t≤1)=0.TofindthedistributionoftheminimumofBrownianmotionm(t)=min0≤s≤tB(s)weusethesymmetryargument,andthat−minB(s)=max(−B(s)).0≤s≤t0≤s≤t 3.7.DISTRIBUTIONOFHITTINGTIMES73Theorem3.16IfB(t)isaBrownianMotionwithB(0)=0,thenBˆ(t)=−B(t)isalsoaBrownianmotionwithBˆ(0)=0.Proof:TheprocessBˆ(t)=−B(t)hasindependentandnormallydistributedincrements.Italsohascontinuouspaths,thereforeitisBrownianmotion.
Theorem3.17Foranyx<0P0(minB(s)≤x)=2P0(B(t)≥−x)=2P0(B(t)≤x)0≤s≤tTheproofisstraightforwardandisleftasanexercise.3.7DistributionofHittingTimesTxisfinitebyTheorem3.14.TheresultbelowgivesthedistributionofTxandestablishesthatTxhasinfinitemean.Theorem3.18TheprobabilitydensityofTxisgivenby2|x|−3−xfTx(t)=√t2e2t,2π2whichistheInverseGammadensitywithparameters1andx.ET=+∞.220xProof:Takex>0.Theevents{M(t)≥x}and{Tx≤t}arethesame,sothatP(Tx≤t)=P(M(t)≥x)+∞22−y=2P(B(t)≥x)=e2tdy.xπtTheformulaforthedensityofTxisobtainedbydifferentiationafterthechangeyofvariablesu=√intheintegral.Finally,t∞|x|−1−x2−1−x2√E0Tx=√t2e2tdt=∞,sincet2e2t∼1/t,t→∞.2π0Forx<0theproofissimilar.
Remark3.6:ThepropertyP(Tx<∞)=1iscalledtherecurrencepropertyofBrownianmotion.AlthoughP(Tx<∞)=1,E(Tx)=∞,eventhoughxisvisitedwithprobabilityone,theexpectedtimeforittohappenisinfinite. 74CHAPTER3.BASICSTOCHASTICPROCESSESThenextresultlooksathittingtimesTxasaprocessinx.Theorem3.19Theprocessofhittingtimes{Tx},x≥0,hasincrementsindependentofthepast,thatis,forany00andy>x,Bˆ(t)beB(t)reflectedatTy.ThenP(B(t)≤x,M(t)≥y)=P(Ty≤t,B(t)≤x)(since{M(t)≥y}={Ty≤t})=P(Ty≤t,Bˆ(t)≥2y−x)(on{Ty≤t},Bˆ(t)=2y−B(t))=P(Ty≤t,B(t)≥2y−x)(sinceTyisthesameforBandBˆ)=P(B(t)≥2y−x)(sincey−x>0,and{B(t)≥2y−x}⊂{Ty≤t})2y−x=1−Φ(√).tThedensityisobtainedbydifferentiation.
Itispossibletoshow(seeforexample,KaratzasandShreve1988,p.123-24)that|B(t)|andM(t)−B(t)havethesamedistribution.Theorem3.22Thetwoprocesses|B(t)|andM(t)−B(t)arebothMarkovprocesseswithtransitionprobabilitydensityfunctionpt(x,y)+pt(x,−y),where2(y−x)p(x,y)=√1e−2tisthetransitionprobabilityfunctionofBrownianmo-t2πttion.Consequentlytheyhavesamefinite-dimensionaldistributions.ThenextresultgivesthejointdistributionofB(t),M(t)andm(t),foraproofseeFreedman1971,p.26-27.Theorem3.23P(a0anda<00theproofissimilar,andisbasedonthedistributionoftheminimumofBrownianmotion.
UsingthisresultwecanestablishTheorem3.25TheprobabilitythatBrownianmotionB(t)hasatleastonezerointhetimeinterval(a,b)isgivenby+2aarccos.πbProof:Denotebyh(x)=P(Bhasatleastonezeroin(a,b)|Ba=x).BytheMarkovpropertyP(Bhasatleastonezeroin(a,b)|Ba=x)isthesameasP(Bxhasatleastonezeroin(0,b−a)).ByconditioningP(Bhasatleastonezeroin(a,b))∞=P(Bhasatleastonezeroin(a,b)|Ba=x)P(Ba∈dx)−∞+∞∞2x2−=h(x)P(Ba∈dx)=h(x)e2adx.−∞πa0Puttingintheexpressionforh(x)fromthepreviousexampleandperformingthenecessarycalculationsweobtaintheresult.
TheArcsinelawnowfollows: 3.9.ZEROSOFBROWNIANMOTION.ARCSINELAW77Theorem3.26TheprobabilitythatBrownianmotion{B(t)}hasnozerosinthetimeinterval(a,b)isgivenby2arcsina.πbThenextresultgivesdistributionsofthelastzerobeforet,andthefirstzeroaftert.Letγt=sup{s≤t:B(s)=0}=lastzerobeforet.(3.18)βt=inf{s≥t:B(s)=0}=firstzeroaftert.(3.19)Notethatβtisastoppingtimebutγtisnot.Theorem3.27+2xP(γt≤x)=arcsin.(3.20)πt+2tP(βt≥y)=arcsin.(3.21)πy+2xP(γt≤x,βt≥y)=arcsin.(3.22)πyProof:Allofthesefollowfromthepreviousresult.Forexample,P(γt≤x)=P(Bhasnozerosin(x,t)).P(γt≤x,βt≥y)=P(Bhasnozerosin(x,y)).
SinceBrownianmotioniscontinuous,andithasnozerosontheinterval(γt,βt)itkeepsthesamesignonthisinterval,eitherpositiveornegative.WhenBrownianmotionisentirelypositiveorentirelynegativeonaninterval,itissaidthatitisanexcursionofBrownianmotion.Thusthepreviousresultstatesthatexcursionshavethearcsinelaw.TopictureaBrownianpathconsiderforeveryrealizationB={B(t),0≤t≤1},thesetofitszerosontheinterval[0,1],thatis,therandomsetL0=L0(B)={t:B(t)=0,0≤t≤1}.Theorem3.28ThesetofzerosofBrownianmotionisarandomuncountableclosedsetwithoutisolatedpointsandhasLebesguemeasurezero.Proof:AccordingtotheExample3.8,theprobabilitythatBrownianmotionstaysbelowzeroontheinterval[0,1]iszero.Thereforeitchangessignonthisinterval.Thisimplies,sinceBrownianmotioniscontinuous,thatithasazeroinside[0,1].Thesamereasoningleadstotheconclusionthatforanypositivet,theprobabilitythatBrownianmotionhasthesamesignontheinterval[0,t]iszero.Thereforeithasazeroinside[0,t]foranyt,nomatterhowsmallitis.Thisimpliesthatthesetofzerosisaninfiniteset,moreovertimet=0isalimitofzerosfromtheright. 78CHAPTER3.BASICSTOCHASTICPROCESSESObservenext,thatthesetofzerosisclosed,thatis,ifB(τn)=0andlimn→∞τn=τthenB(τ)=0.ThisistruesinceB(t)isacontinuousfunctionoft.ByusingthestrongMarkovproperty,itispossibletoseethatanyzeroofBrownianmotionisalimitofotherzeros.IfB(τ)=0,andτisastoppingtime,thenby(3.13)Bˆ(t)=B(t+τ)−B(τ)=B(t+τ)isagainBrownianmotionstartedanewattimeτ.Thereforetimet=0forthenewBrownianmotionBˆisalimitfromtherightofzerosofBˆ.ButBˆ(t)=B(t+τ),sothatτalimitfromtherightofzerosofB.However,noteveryzeroofBrownianmotionisastoppingtime.Forexample,forafixedt,γt,thelastzerobeforetisnotastoppingtime.Nevertheless,usingamoreintricateargument,onecanseethatanyzeroisalimitofotherzeros.Asketchisgivenbelow.Ifτisthefirstzeroaftert,thenτisastoppingtime.Thusthesetofallsamplepathssuchthatτisalimitpointofzerosfromtherighthasprobabilityone.Theintersectionofsuchsetsoverallrationalt’sisagainasetofprobabilityone.Thereforeforalmostallsamplepathsthefirstzerothatfollowsanyrationalnumberisalimitofzerosfromtheright.ThisimpliesthatanypointofL0isalimitofpointsfromL0(itisaperfectset).Ageneralresultfromthesettheory,whichisnothardtoprove,statesthatifaninfinitesetcoincideswiththesetofitslimitpoints,thenitisuncountable.Althoughuncountable,L0hasLebesguemeasurezero.Thisisseenby1writingtheLebesguemeasureofL0as|L0|=0I(B(t)=0)dt.Itisanon-negativerandomvariable.Takingtheexpectation,andinterchangingtheintegralsbyFubini’stheorem11E|L0|=EI(B(t)=0)dt=P(B(t)=0)dt=0.00ThisimpliesP(|L0|=0)=1.
Theorem3.29AnylevelsetLa={t:B(t)=a,0≤t≤1}hasthesamepropertiesasL0.Proof:LetTabethefirsttimewithB(t)=a.ThenbythestrongMarkovproperty,Bˆ(t)=BTa+t−BTa=BTa+t−aisaBrownianmotion.ThesetofzerosofBˆisthelevelasetofB.
3.10SizeofIncrementsofBrownianMotionIncrementsoverlargetimeintervalssatisfytheLawofLargeNumbersandtheLawoftheIteratedLogarithm.Forproofssee,forexample,KaratzasandShreve(1988). 3.10.SIZEOFINCREMENTSOFBROWNIANMOTION79Theorem3.30(LawofLargeNumbers)B(t)lim=0a.s.t→∞tAmorepreciseresultisprovidedbytheLawofIteratedLogarithm.Theorem3.31(LawofIteratedLogarithm)B(t)limsup√=1,a.s.t→∞2tlnlntB(t)liminf√=−1a.s.t→∞2tlnlntToobtainthebehaviourforsmalltnearzerotheprocessW(t)=tB(1/t)isconsidered,whichisalsoBrownianmotion.Example3.9:LetB(t)beBrownianmotion.TheprocessW(t)definedasW(t)=tB(1/t),fort>0,andW(0)=0,isalsoBrownianmotion.Indeed,W(t)hascontinuouspaths.ContinuityatzerofollowsfromtheLawofLargeNumbers.Itis,clearly,aGaussianprocess,andhaszeromean.ItscovarianceisgivenbyCov(W(t),W(s))=E(W(t)W(s))=tsE(B(1/t)B(1/s))=ts(1/t)=s,forsn}∈Fn.TakeHn=1ifn≤τ,andHn=0ifn>τ,inotherwords,Hn=I(τ≥n).ThenHnispredictable,because{τ≥n}={τ>n+1}∈Fn+1.Thestochasticintegralgivesthemartingalestoppedatτ,(H·M)n=H0M0+H1(M1−M0)+···+Hn(Mn−Mn−1)=Mτ∧n=MτI(τ≤n)+MnI(τ>n).SinceHn=I(τ≥n)isboundedby1,Theorem3.37impliesthattheprocess(H·M)n=Mτ∧nisamartingale.ThuswehaveshownTheorem3.39Amartingalestoppedatastoppingtimeτ,Mτ∧nisamar-tingale.Inparticular,EMτ∧n=EM0.(3.29)CommentherethatTheorem3.39holdsalsoincontinuoustime,seeTheorem7.14.ItisaBasicStoppingresult,whichishardertoprove.Example3.10:(Doublingbetsstrategy).ConsiderthedoublingstrategywhenbettingontheHeadsintossesofafaircoin.BetH1=1.IfHeadscomesupthenstop.TheprofitisG1=1.IftheoutcomeisTails,thenbetH2=2onthesecondtoss.Ifthesecondtosscomesupheads,thenstop.TheprofitisG2=4−3=1.Ifthegamecontinuesfornsteps,(meaningthatn−1then−1tossesdidnotresultinawin)thenbetHn=2onthen-thtoss.Ifthen−1n−1n-thtosscomesupHeadsthenstop.TheprofitisGn=2×2−(1+2+...2)=nn2−(2−1)=1.TheprobabilitythatthegamewillstopatafinitenumberofstepsisoneminustheprobabilitythatHeadsnevercomeup.ProbabilityofonlyTails−nonthefirstntossesis,byindependence,2.TheprobabilitythatHeadsnever−ncomesupisthelimitlimn→∞2=0,thusthegamewillstopforsure.Foranynon-randomtimeTthegainprocessGt,t≤Tisamartingalewithzeromean.Thedoublingstrategydoesnotcontradicttheresultabove,becausethestrategyusesanunboundedstoppingtime,thefirsttimeonedollariswon.FurtherinformationondiscretetimemartingalesandontheirstoppingisgiveninChapter7. 86CHAPTER3.BASICSTOCHASTICPROCESSES3.14PoissonProcessIfBrownianmotionprocessisabasicmodelforcumulativesmallnoisepresentcontinuously,thePoissonprocessisabasicmodelforcumulativenoisethatoccursasashock.Letλ>0.ArandomvariableXhasaPoissondistributionwithpa-rameterλ,denotedPn(λ),ifittakesnon-negativeintegervaluesk≥0withprobabilitiesλkP(X=k)=e−λ,k=0,1,2,......(3.30)k!ThemomentgeneratingfunctionofthisdistributionisgivenbyuE(euX)=eλ(e−1).(3.31)DefiningPropertiesofPoissonprocessAPoissonprocessN(t)isastochasticprocesswiththefollowingproperties.1.(Independenceofincrements)N(t)−N(s)isindependentofthepast,thatis,ofFs,theσ-fieldgeneratedbyN(u),u≤s.2.(Poissonincrements)N(t)−N(s),t>s,hasaPoissondistributionwithparameterλ(t−s).IfN(0)=0,thenN(t)hasthePn(λt)distribution.3.(Stepfunctionpaths)ThepathsN(t),t≥0,areincreasingfunctionsoftchangingonlybyjumpsofofsize1.Remark3.9:AdefinitionofaPoissonprocessinamoregeneralmodel(thatcontainsextrainformation)isgivenbyapair{N(t),Ft},t≥0,whereFtisanincreasingsequenceofσ-fields(afiltration),N(t)isanadaptedprocess,i.e.N(t)isFtmeasurable,suchthatProperties1-3abovehold.Consideramodelforoccurrenceofindependentevents.Definetherateλastheaveragenumberofeventsperunitoftime.LetN(t)bethenumberofeventsthatoccuruptotimet,i.e.inthetimeinterval(0,t].ThenN(t)−N(s)givesthenumberofeventsthatoccurinthetimeinterval(s,t].APoissonprocessN(t)canbeconstructedasfollows.Letτ1,τ2,....beindependentrandomvariableswiththeexponentialexp(λ)distribution,thatis,P(τ>t)=e−λt.τ’srepresentthetimesbetweenoccurrenceofsuccessive1nevents.LetTn=i=1τi,bethetimeofthen-thevent.ThenN(t)=sup{n:Tn≤t} 3.14.POISSONPROCESS87countsthenumberofeventsuptotimet.ItisnothardtoverifythedefiningpropertiesofthePoissonprocessforthisconstruction.N(t)hasthePoissondistributionwithparameterλt.Consequently,(λt)kP(N(t)=k)=e−λt,k=0,1,2,.....,k!EN(t)=λt,andVar(N(t))=λt.VariationandQuadraticVariationofthePoissonProcessLet0=tnn).n=0Exercise3.4:LetB(t)beaBrownianmotion.ShowthatthefollowingprocessesareBrownianmotionson[0,T].1.X(t)=−B(t).2.X(t)=B(T−t)−B(T),whereT<+∞.3.X(t)=cB(t/c2),whereT≤+∞.4.X(t)=tB(1/t),t>0,andX(0)=0.Hint:Checkthedefiningproperties.Alternatively,showthattheprocessisaGaussianprocesswithcorrelationfunctionmin(s,t).Alternatively,showthattheprocessisacontinuousmartingalewithquadraticvariationt(thisistheLevy’scharacterization,andwillbeprovenlater.) 3.15.EXERCISES89Exercise3.5:LetB(t)andW(t)betwoindependentBrownianmotions.√ShowthatX(t)=(B(t)+W(t))/2isalsoaBrownianmotion.Findcorre-lationbetweenB(t)andX(t).Exercise3.6:LetB(t)beann-dimensionalBrownianmotion,andxisanon-randomvectorinIRnwithlength1,|x|2=1.ShowthatW(t)=x·B(t)isa(one-dimensional)Brownianmotion.Exercise3.7:LetB(t)beaBrownianmotionand0≤t1,...≤tn.GiveamatrixA,suchthat(B(t),B(t),...,B(t))T=A(Z,...,Z)T,where12n1nZi’sarestandardNormalvariables.Hencegivethecovariancematrixof(B(t),B(t),...,B(t)).HereTstandsfortransposeandthevectorsare12ncolumnvectors.Exercise3.8:LetB(t)beaBrownianmotionand0≤s0,E(Z(t)Z(t+s))=0)suchthat
pX(t)=asX(t−s)+Z(t).s=11.ShowthatX(t)isMarkovianifandonlyifp=1.2.ShowthatifX(t)isAR(2),thenY(t)=(X(t),X(t+1))isMarkovian.3.SupposethatZ(t)isaGaussianprocess.WritethetransitionprobabilityfunctionofanAR(1)processX(t).Exercise3.23:Thedistributionofarandomvariableτhasthelackofmem-orypropertyifP(τ>a+b|τ>a)=P(τ>b).Verifythelackofmemorypropertyfortheexponentialexp(λ)distribution.Showthatifτhasthelackofmemorypropertyandadensity,thenithasanexponentialdistribution. Chapter4BrownianMotionCalculusInthischapterstochasticintegralswithrespecttoBrownianmotionareintro-ducedandtheirpropertiesaregiven.TheyarealsocalledItˆointegrals,andthecorrespondingcalculusItˆocalculus.4.1DefinitionofItˆoIntegralTOurgoalistodefinethestochasticintegralX(t)dB(t),alsodenotedXdB0orX·B.ThisintegralshouldhavethepropertythatifX(t)=1thenTdB(t)=B(T)−B(0).Similarly,ifX(t)isaconstantc,thentheintegral0shouldbec(B(T)−B(0)).InthiswaywecanintegrateconstantprocesseswithrespecttoB.Theintegralover(0,T]shouldbethesumofintegralsovertwosubintervals(0,a]and(a,T].ThusifX(t)takestwovaluesc1on(0,a],andc2on(a,T],thentheintegralofXwithrespecttoBiseasilydefined.Inthiswaytheintegralisdefinedforsimpleprocesses,thatis,processeswhichareconstantonfinitelymanyintervals.Bythelimitingproceduretheintegralisthendefinedformoregeneralprocesses.ItˆoIntegralofSimpleProcessesConsiderfirstintegralsofanon-randomsimpleprocessX(t),whichisafunc-tionoftanddoesnotdependonB(t).Bydefinitionasimplenon-randomprocessX(t)isaprocessforwhichthereexisttimes0=t0ti.IfFtistheσ-fieldgeneratedbyBrownianmotionuptotimet,thenξiisFti-measurable.TheapproachofdefiningtheintegralbyapproximationcanbecarriedoutfortheclassofadaptedprocessesX(t),0≤t≤T.Definition4.1AprocessXiscalledadaptedtothefiltrationIF=(Ft),ifforallt,X(t)isFt-measurable.Remark4.1:Inorderthattheintegralhasdesirableproperties,inpartic-ularthattheexpectationandtheintegralcanbeinterchanged(byFubini’stheorem),therequirementthatXisadaptedistooweak,andastrongercon-dition,thatofaprogressive(progressivelymeasurable)processisneeded.X 4.1.DEFINITIONOFITOINTEGRALˆ93isprogressiveifitisameasurablefunctioninthepairofvariables(t,ω),i.e.B([0,t])×Ftmeasurableasamapfrom[0,t]×ΩintoIR.Itcanbeseenthateveryadaptedright-continuouswithleftlimitsorleft-continuouswithrightlimits(regular,cadlag)processisprogressive.Sinceitiseasiertounder-standwhatismeantbyaregularadaptedprocess,weuse‘regularadapted’terminologywithoutfurtherreferencetoprogressiveormeasurablein(t,ω)processes.Definition4.2AprocessX={X(t),0≤t≤T}iscalledasimpleadaptedprocessifthereexisttimes0=t0t),andE(ξ2)<∞,iiii=0,...,n−1;suchthatn
−1X(t)=ξ0I0(t)+ξiI(ti,ti+1](t).(4.3)i=0TForsimpleadaptedprocessesItˆointegralXdBisdefinedasasum0Tn
−1X(t)dB(t)=ξiB(ti+1)−B(ti).(4.4)0i=0Notethatwhenξi’sarerandom,theintegralneednothaveaNormaldistri-bution,asinthecaseofnon-randomci’s.Remark4.2:Simpleadaptedprocessesaredefinedasleft-continuousstepfunctions.Onecantakeright-continuousfunctions.However,whenthestochas-ticintegralisdefinedwithrespecttogeneralmartingales,otherthantheBrow-nianmotion,onlyleft-continuousfunctionsaretaken.PropertiesoftheItˆoIntegralofSimpleAdaptedProcessesHereweestablishmainpropertiesoftheItˆointegralofsimpleprocesses.ThesepropertiescarryovertotheItˆointegralofgeneralprocesses.1.Linearity.IfX(t)andY(t)aresimpleprocessesandαandβaresomeconstantsthenTTT(αX(t)+βY(t))dB(t)=αX(t)dB(t)+βY(t)dB(t).0002.FortheindicatorfunctionofanintervalI(a,b](t)(I(a,b](t)=1whent∈(a,b],andzerootherwise)TTbI(a,b](t)dB(t)=B(b)−B(a),I(a,b](t)X(t)dB(t)=X(t)dB(t).00a 94CHAPTER4.BROWNIANMOTIONCALCULUST3.Zeromeanproperty.EX(t)dB(t)=004.Isometryproperty.2TT2EX(t)dB(t)=E(X(t))dt(4.5)00Proof:Properties1and2areverifieddirectlyfromthedefinition.Proofoflinearityoftheintegralfollowsfromthefactthatalinearcombinationofsimpleprocessesisagainasimpleprocess,andsoisI(a,b](t)X(t).Sinceξi’saresquareintegrable,thenbytheCauchy-Schwarzinequality'',E'ξi(B(ti+1)−B(ti))'≤E(ξi2)E(B(ti+1)−B(ti))2<∞,whichimpliesthat'n
−1'n
−1''E'ξiB(ti+1)−B(ti)'≤E'ξiB(ti+1)−B(ti)'<∞,(4.6)i=0i=0andthestochasticintegralhasexpectation.BythemartingalepropertyofBrownianmotion,usingthatξi’sareFti-measurableEξi(B(ti+1)−B(ti))|Fti=ξiE(B(ti+1)−B(ti))|Fti=0,(4.7)anditfollowsthatEξi(B(ti+1)−B(ti))=0,whichimpliesProperty3.ToproveProperty4,writethesquareasthedoublesum2n
−1n
−122Eξi(B(ti+1)−B(ti))=Eξi(B(ti+1)−B(ti))i=0i=0
+2Eξiξj(B(ti+1)−B(ti))(B(tj+1)−B(tj)).(4.8)i0,Ef2(B(s))ds≤KT.TheresultfollowsbyTheorem4.7.0
QuadraticVariationandCovariationofItˆoIntegralstTheItˆointegralY(t)=X(s)dB(s),0≤t≤T,isarandomfunctionoft.0Itiscontinuousandadapted.ThequadraticvariationofYisdefinedby(see(1.13))n
−1[Y,Y](t)=lim(Y(tn)−Y(tn))2,(4.22)i+1ii=0whereforeachn,{tn}n,isapartitionof[0,t],andthelimitisinprobability,ii=0takenoverallpartitionswithδ=max(tn−tn)→0asn→∞.nii+1itTheorem4.9ThequadraticVariationoftheItˆointegralX(s)dB(s)is0givenby"tt#tX(s)dB(s),X(s)dB(s)(t)=X2(s)ds.(4.23)000Itiseasytoverifytheresultforsimpleprocesses,seetheExamplebelow.Thegeneralcasecanbeprovedbyapproximationsbysimpleprocesses.Example4.8:Forsimplicity,supposethatXtakesonlytwodifferentvalueson[0,1]:ξ0on[0,1/2]andξ1on[1/2,1]Xt=ξ0I[0,1/2](t)+ξ1I(1/2,1](t). 102CHAPTER4.BROWNIANMOTIONCALCULUSItiseasytoseethat$tξ0B(t)ift≤1/2Y(t)=X(s)ds=ξ0B(1/2)+ξ1B(t)−B(1/2)ift>1/2.0Thusforanypartitionof[0,t],$nnnnY(tn)−Y(tn)=ξ0B(ti+1)−B(ti)ifti1/2
n−12[Y,Y](t)=lim(Y(ti+1)−Y(ti))i=0

2222=ξ0lim(B(ti+1)−B(ti))+ξ1lim(B(ti+1)−B(ti))ti<1/2ti>1/2t222=ξ0[B,B](1/2)+ξ1[B,B]((1/2,t])=X(s)ds.0nnThelimitsabovearelimitsinprobabilitywhenδn=maxi{(ti+1−ti)}→0.Inthesameway(4.23)isverifiedforanysimplefunction.Example4.9:Usingtheformula(4.23),quadraticvariationoftheItˆointegral"#·t2B(s)dB(s)(t)=B(s)ds.00tCorollary4.10IfX2(s)ds>0,forallt≤T,thentheItˆointegral0tY(t)=X(s)dB(s)hasinfinitevariationon[0,t]forallt≤T.0Proof:IfY(t)wereoffinitevariation,itsquadraticvariationwouldbezero,leadingtoacontradiction.
LikeBrownianmotion,theItˆointegralY(t)isacontinuousbutnowheredif-ferentiablefunctionoft. 4.3.ITOINTEGRALANDGAUSSIANPROCESSESˆ103LetnowY1(t)andY2(t)beItˆointegralsofX1(t)andX2(t)withrespecttothesameBrownianmotionB(t).Then,clearly,theprocessY1(t)+Y2(t)isalsoanItˆointegralofX1(t)+X2(t)withrespecttoB(t).QuadraticcovariationofY1andY2on[0,t]isdefinedby1[Y1,Y2](t)=([Y1+Y2,Y1+Y2](t)−[Y1,Y1](t)−[Y2,Y2](t)).(4.24)2By(4.23)itfollowsthatt[Y1,Y2](t)=X1(s)X2(s)ds.(4.25)0Itisclearthat[Y1,Y2](t)=[Y2,Y1](t),anditcanbeseenthatquadraticcovariationisgivenbythelimitinprobabilityofproductsofincrementsoftheprocessesYandYwhenpartitions{tn}of[0,t]shrink,12in
−1[Y,Y](t)=limY(tn)−Y(tn)Y(tn)−Y(tn).121i+11i2i+12ii=04.3ItˆoIntegralandGaussianProcessesWehaveseeninSection4.1thattheItˆointegralofsimplenon-randompro-cessesisaNormalrandomvariable.Itiseasytoseebyusingmomentgener-atingfunctions(seeExercise4.3)thatalimitinprobabilityofsuchasequenceisalsoGaussian.Thisimpliesthefollowingresult.TTheorem4.11IfX(t)isnon-randomsuchthatX2(s)ds<∞,thenits0tItˆointegralY(t)=X(s)dB(s)isaGaussianprocesswithzeromeanand0covariancefunctiongivenbytCov(Y(t),Y(t+u))=X2(s)ds,u≥0.(4.26)0Moreover,Y(t)isasquareintegrablemartingale.t2t2Proof:Sincetheintegrandisnon-random,EX(s)ds=X(s)ds<∞.00BythezeromeanpropertyofItˆointegral,Yhaszeromean.Tocomputethet+utt+ucovariancefunction,writeas+andusethemartingaleproperty00tofY(t)toobtain)t)t+u'**'EX(s)dB(s)EX(s)dB(s)'Ft=0.0t 104CHAPTER4.BROWNIANMOTIONCALCULUSHence)tt+u*Cov(Y(t),Y(t+u))=EX(s)dB(s)X(s)dB(s)00)t*2tt=EX(s)dB(s)=EX2(s)ds=X2(s)ds.000
Aproofofnormalityofintegralsofnon-randomprocesseswillbedonelaterbyusingItˆo’sformula.t3Example4.10:AccordingtoTheorem4.11J=sdB(s)hasaNormalN(0,t/3)0distribution.Example4.11:4LetX(t)=2I[0,1](t)+3I(1,3](t)−5I(3,4](t).GivetheItˆointegralX(t)dB(t)as0asumofrandomvariables,giveitsdistribution,meanandvariance.ShowthatthetprocessM(t)=X(s)dB(s),0≤t≤4,isaGaussianprocessandamartingale.04134X(t)dB(t)=X(t)dB(t)+X(t)dB(t)+X(t)dB(t)0013134=2dB(t)+3dB(t)+(−5)dB(t)013=2(B(1)−B(0))+3(B(3)−B(1))−5(B(4)−B(3)).TheItˆointegralisasumof3independentNormalrandomvariables(byindependenceofincrementsofBrownianmotion),2N(0,1)+3N(0,2)−5N(0,1).ItsdistributionisN(0,47).tThemartingalepropertyandtheGaussianpropertyofM(t)=X(s)dB(s),00≤t≤4,followfromtheindependenceoftheincrementsofM(t),zeromeanincre-tmentsandtheNormalityoftheincrements.M(t)−M(s)=X(u)dB(u).Takestforexample00tt+uY(t+u)=X(t+u,s)dB(s)+X(t+u,s)dB(s).0tt+uSinceX(t+u,s)isnon-random,theItˆointegralX(t+u,s)dB(s)istindependentofFt.Therefore)tt+u*EX(t,s)dB(s)X(t+u,s)dB(s)=0,0tandCov(Y(t),Y(t+u))=E(Y(t)Y(t+u)))tt*=EX(t,s)dB(s)X(t+u,s)dB(s)00t=X(t,s)X(t+u,s)ds,(4.28)0wherethelastequalityisobtainedbytheexpectationofaproductofItˆointegrals,Equation(4.19).
4.4Itˆo’sFormulaforBrownianMotionItˆo’sformula,alsoknownasthechangeofvariableandthechainrule,isoneofthemaintoolsofstochasticcalculus.Itgivesrisetomanyothers,suchasDynkin,Feynman-Kac,andintegrationbypartsformulae.Theorem4.13IfB(t)isaBrownianmotionon[0,T]andf(x)isatwicecontinuouslydifferentiablefunctiononIR,thenforanyt≤Ttt1f(B(t))=f(0)+f(B(s))dB(s)+f(B(s))ds.(4.29)020 106CHAPTER4.BROWNIANMOTIONCALCULUSProof:Notefirstthatbothintegralsin(4.29)arewelldefined,theItˆointegralbyCorollary4.4.Let{tn}beapartitionof[0,t].Clearly,in
−1f(B(t))=f(0)+f(B(tn))−f(B(tn)).i+1ii=0ApplyTaylor’sformulatof(B(tn))−f(B(tn))toobtaini+1innnnn1nnn2f(B(ti+1))−f(B(ti))=f(B(ti))(B(ti+1)−B(ti))+f(θi)(B(ti+1)−B(ti)),2whereθn∈(B(tn),B(tn)).Thus,iii+1n
−1f(B(t))=f(0)+f(B(tn))(B(tn)−B(tn))ii+1ii=0n
−11nnn2+f(θi)(B(ti+1)−B(ti)).(4.30)2i=0Takinglimitsasδn→0,thefirstsumin(4.30)convergestotheItˆointegraltf(B(s))dB(s).Bythetheorembelowthesecondsumin(4.30)converges0ttof(B(s))dsandtheresultfollows.0
Theorem4.14Ifgisaboundedcontinuousfunctionand{tn}representsipartitionsof[0,t],thenforanyθn∈(B(tn),B(tn)),thelimitinprobabilityiii+1n
−1tnnn2limg(θi)B(ti+1)−B(ti)=g(B(s))ds.(4.31)δn→00i=0Proof:Takefirstθn=B(tn)tobetheleftendoftheinterval(B(tn),B(tn)).iiii+1Weshowthatthesumsconvergeinprobabilityn
−1tnnn2g(B(ti))B(ti+1)−B(ti)→g(B(s))ds.(4.32)i=00Bycontinuityofg(B(t))anddefinitionoftheintegral,itfollowsthatn
−1tg(B(tn))(tn−tn)→g(B(s))ds.(4.33)ii+1ii=00NextweshowthatthedifferencebetweenthesumsconvergestozeroinL2,n
−1n
−1nnn2g(B(ti))B(ti+1)−B(ti)−g(B(ti))(ti+1−ti)→0.(4.34)i=0i=0 4.4.ITO’SFORMULAFORBROWNIANMOTIONˆ107With∆B=B(tn)−B(tn)and∆t=tn−tn,byusingconditioningitisii+1iii+1iseenthatthecross-productterminthefollowingexpressionvanishesand2n
−1n
−1n22n22Eg(B(ti))(∆Bi)−∆ti=Eg(B(ti))E(∆Bi)−∆ti|Ftii=0i=0n
−1n
−1=2Eg2(B(tn))(∆t)2≤δ2Eg2(B(tn))∆t→0asδ→0.iiiii=0i=0Itfollowsthatn
−1g(B(tn))(∆B)2−∆t→0,iiii=0inthesquaremean(L2),implying(4.34)andthatbothsumsin(4.33)and(4.32)havethesamelimit,and(4.32)isestablished.Nowforanychoiceofθn,wehaveasδ→0,inn
−1nnnn2g(θi)−g(B(ti))B(ti+1)−B(ti)i=0n
−1nnnn2≤maxg(θi)−g(B(ti))B(ti+1)−B(ti)→0.(4.35)ii=0ThefirsttermconvergestozeroalmostsurelybycontinuityofgandB,andthesecondconvergesinprobabilitytothequadraticvariationofBrownianmotion,t,implyingconvergencetozeroinprobabilityin(4.35).Thisimpliesn−1n2n−12thatbothsumsi=0g(θi)(∆Bi)andi=0g(B(ti))(∆Bi)havethesamelimitinprobability,andtheresultfollowsby(4.32).
mExample4.12:Takingf(x)=x,m≥2,wehavettmm−1m(m−1)m−2B(t)=mB(s)dB(s)+B(s)ds.200Withm=2,t2B(t)=2B(s)dB(s)+t.0Rearranging,werecovertheresultonthestochasticintegralt121B(s)dB(s)=B(t)−t.220xExample4.13:Takingf(x)=e,wehavettB(t)B(s)1B(s)e=1+edB(s)+eds.200 108CHAPTER4.BROWNIANMOTIONCALCULUS4.5ItˆoProcessesandStochasticDifferentialsDefinitionofItˆoProcessesAnItˆoprocesshastheformttY(t)=Y(0)+µ(s)ds+σ(s)dB(s),0≤t≤T,(4.36)00whereY(0)isF0-measurable,processesµ(t)andσ(t)areFt-adapted,suchTT2that|µ(t)|dt<∞andσ(t)dt<∞.00ItissaidthattheprocessY(t)hasthestochasticdifferentialon[0,T]dY(t)=µ(t)dt+σ(t)dB(t),0≤t≤T.(4.37)Weemphasizethatarepresentation(4.37)onlyhasmeaningbythewayof(4.36),andnoother.Notethattheprocessesµandσin(4.36)may(andoftendo)dependonY(t)orB(t)aswell,orevenonthewholepastpathofB(s),s≤t;forexampletheymaydependonthemaximumofBrownianmotionmaxs≤tB(s).Example4.14:Example4.12showsthatt2B(t)=t+2B(s)dB(s).(4.38)02ttInotherwords,withY(t)=B(t)wecanwriteY(t)=ds+2B(s)dB(s).Thus002µ(s)=1andσ(s)=2B(s).ThestochasticdifferentialofB(t)2d(B(t))=2B(t)dB(t)+dt.Theonlymeaningthishasistheintegralrelation(4.38).B(t)Example4.15:Example4.13showsthatY(t)=ehasstochasticdifferentialB(t)B(t)1B(t)de=edB(t)+edt,2or1dY(t)=Y(t)dB(t)+Y(t)dt.2Itˆo’sformula(4.29)indifferentialnotationbecomes:foraC2functionf1d(f(B(t))=f(B(t))dB(t)+f(B(t))dt.(4.39)2 4.5.ITOPROCESSESANDSTOCHASTICDIFFERENTIALSˆ109Example4.16:Wefindd(sin(B(t))).f(x)=sin(x),f(x)=cos(x),f(x)=−sin(x).Thus1d(sin(B(t)))=cos(B(t))dB(t)−sin(B(t))dt.2Similarly,1d(cos(B(t)))=−sin(B(t))dB(t)−cos(B(t))dt.2iB(t)2Example4.17:Wefindd(e)withi=−1.TheapplicationofItˆo’sformulatoacomplex-valuedfunctionmeansitsapplicationtotherealandcomplexpartsofthefunction.Aformalapplicationbytreatingiasanotherconstantgivesthesameresult.Usingtheaboveexample,wecancalculateiB(t)d(e)=dcos(B(t))+idsin(B(t)),ordirectlybyusingItˆo’sformulawithixixixf(x)=e,wehavef(x)=ie,f(x)=−eand1iB(t)iB(t)iB(t)de=iedB(t)−edt.2iB(t)ThusX(t)=ehasstochasticdifferential1dX(t)=iX(t)dB(t)−X(t)dt.2QuadraticVariationofItˆoProcessesLetY(t)beanItˆoprocessttY(t)=Y(0)+µ(s)ds+σ(s)dB(s),(4.40)00whereitisassumedthatµandσaresuchthattheintegralsinquestionaredefined.Thenbythepropertiesoftheintegrals,Y(t),0≤t≤T,isa(random)tcontinuousfunction,theintegralµ(s)dsisacontinuousfunctionoftandis0offinitevariation(itisdifferentiablealmosteverywhere),andtheItˆointegraltσ(s)dB(s)iscontinuous.QuadraticvariationofYon[0,t]isdefinedby0(see(1.13))n
−1nn2[Y](t)=[Y,Y]([0,t])=limY(ti+1)−Y(ti),(4.41)δn→0i=0whereforeachn,{tn},isapartitionof[0,t],andthelimitisinprobabilityitakenoverpartitionswithδ=max(tn−tn)→0asn→∞,andisgivennii+1iby"··#[Y](t)=µ(s)ds+σ(s)dB(s)(t)00"·#"··#"·#=µ(s)ds(t)+2µ(s)ds,σ(s)dB(s)(t)+σ(s)dB(s)(t).0000 110CHAPTER4.BROWNIANMOTIONCALCULUSTheresultonthecovariation,Theorem1.11,statesthatthequadraticcovariationofacontinuousfunctionwithafunctionoffinitevariationiszero.tThisimpliesthatthequadraticcovariationoftheintegralµ(s)dswithterms0aboveiszero,andweobtainbyusingtheresultonthequadraticvariationofItˆointegrals(Theorem4.9)"·#t[Y](t)=σ(s)dB(s)(t)=σ2(s)ds.(4.42)00IfY(t)andX(t)havestochasticdifferentialswithrespecttothesameBrown-ianmotionB(t),thenclearlyprocessY(t)+X(t)alsohasastochasticdiffer-entialwithrespecttothesameBrownianmotion.ItfollowsthatcovariationofXandYon[0,t]existsandisgivenby1[X,Y](t)=[X+Y,X+Y](t)−[X,X](t)−[Y,Y](t).(4.43)2Theorem1.11hasanimportantcorollaryTheorem4.15IfXandYareItˆoprocessesandXisoffinitevariation,thencovariation[X,Y](t)=0.Example4.18:LetX(t)=exp(t),Y(t)=B(t),then[X,Y](t)=[exp,B](t)=0.Introduceaconventionthatallowsaformalmanipulationwithstochasticdif-ferentials.dY(t)dX(t)=d[X,Y](t),(4.44)andinparticular2(dY(t))=d[Y,Y](t).(4.45)SinceX(t)=tisacontinuousfunctionoffinitevariationandY(t)=B(t)iscontinuouswithquadraticvariationt,thefollowingrulesfollow2dB(t)dt=0,(dt)=0,(4.46)but2(dB(t))=d[B,B](t)=dt.(4.47)Remark4.7:Insometexts,forexample,Protter(1992),quadraticvariationisdefinedbyaddingthevalueY2(0)to(4.41).Thedefinitiongivenheregivesamorefamiliarlookingformulaforintegrationbyparts,anditisusedinmanytexts,forexample,RogersandWilliams(1987)p.59,Metivier(1982)p.175. 4.6.ITO’SFORMULAFORITˆOPROCESSESˆ111IntegralswithrespecttoItˆoprocessesItisnecessarytoextendintegrationwithrespecttoprocessesobtainedfromtBrownianmotion.LettheItˆointegralprocessY(t)=X(s)dB(s)bedefined0T2forallt≤T,whereX(t)isanadaptedprocess,suchthatX(s)ds<∞0T22withprobabilityone.LetanadaptedprocessH(t)satisfyH(s)X(s)ds<0t∞withprobabilityone.ThentheItˆointegralprocessZ(t)=H(s)X(s)dB(s)0isalsodefinedforallt≤T.InthiscaseonecanformallywritebyidentifyingdY(t)andX(t)dB(t),ttZ(t)=H(s)dY(s):=H(s)X(s)dB(s).(4.48)00InChapter8integralswithrespecttoY(t)willbeintroducedinadirectway,buttheresultagreeswiththeoneabove.Moregenerally,ifYisanItˆoprocesssatisfyingdY(t)=µ(t)dt+σ(t)dB(t),(4.49)t22tandHisadaptedandsatisfiesH(s)σ(s)ds<∞,|H(s)µ(s)|ds<∞,00tthenZ(t)=H(s)dY(s)isdefinedas0tttZ(t)=H(s)dY(s):=H(s)µ(s)ds+H(s)σ(s)dB(s).(4.50)000Example4.19:Ifa(t)denotesthenumberofsharesheldattimet,thenthegainTfromtradinginsharesduringthetimeinterval[0,T]isgivenbya(t)dS(t).04.6Itˆo’sFormulaforItˆoprocessesTheorem4.16(Itˆo’sformulaforf(X(t)))LetX(t)haveastochasticdif-ferentialfor0≤t≤TdX(t)=µ(t)dt+σ(t)dB(t).(4.51)Iff(x)istwicecontinuouslydifferentiable(C2function),thenthestochasticdifferentialoftheprocessY(t)=f(X(t))existsandisgivenby1df(X(t))=f(X(t))dX(t)+f(X(t))d[X,X](t)212=f(X(t))dX(t)+f(X(t))σ(t)dt(4.52)2)*12=f(X(t))µ(t)+f(X(t))σ(t)dt+f(X(t))σ(t)dB(t).2 112CHAPTER4.BROWNIANMOTIONCALCULUSThemeaningoftheaboveistt12f(X(t))=f(X(0))+f(X(s))dX(s)+f(X(s))σ(s)ds,(4.53)020wherethefirstintegralisanItˆointegralwithrespecttothestochasticdif-ferential.Existenceoftheintegralsintheformula(4.53)isassuredbytheargumentsfollowingTheorem4.13.TheproofalsofollowsthesameideasasTheorem4.13,andisomitted.ProofsofItˆo’sformulacanbefoundinLiptserandShiryaev(2001),p.124,RevuzandYor(2001)p.146,Protter(1992),p.71,RogersandWilliams(1990),p.60.Example4.20:LetX(t)havestochasticdifferential1dX(t)=X(t)dB(t)+X(t)dt.(4.54)2WefindaprocessXsatisfying(4.54).Let’slookforapositiveprocessX.Using2Itˆo’sformulaforlnX(t)((lnx)=1/xand(lnx)=−1/x),112dlnX(t)=dX(t)−2Xtdtbyusingσ(t)=X(t)X(t)2Xt11=dB(t)+dt−dt=dB(t).22SothatlnX(t)=lnX(0)+B(t),andwefindB(t)X(t)=X(0)e.(4.55)UsingItˆo’sformulaweverifythatthisX(t)indeedsatisfies(4.54).Wedon’tclaimatthisstagethat(4.55)istheonlysolution.IntegrationbyPartsWegivearepresentationofthequadraticcovariation[X,Y](t)oftwoItˆopro-cessesX(t)andY(t)intermsofItˆointegrals.Thisrepresentationgivesrisetotheintegrationbypartsformula.Quadraticcovariationisalimitoverdecreasingpartitionsof[0,t],n
−1[X,Y](t)=limX(tn)−X(tn)Y(tn)−Y(tn).(4.56)i+1ii+1iδn→0i=0Thesumontherightabovecanbewrittenasn
−1=X(tn)Y(tn)−X(tn)Y(tn)i+1i+1iii=0 4.6.ITO’SFORMULAFORITˆOPROCESSESˆ113n
−1n
−1nnnnnn−X(ti)Y(ti+1)−Y(ti)−Y(ti)X(ti+1)−X(ti)i=0i=0=X(t)Y(t)−X(0)Y(0)n
−1n
−1nnnnnn−X(ti)Y(ti+1)−Y(ti)−Y(ti)X(ti+1)−X(ti).i=0i=0tThelasttwosumsconvergeinprobabilitytoItˆointegralsX(s)dY(s)and0tY(s)dX(s),cf.Remark(4.4).Thusthefollowingexpressionisobtained0tt[X,Y](t)=X(t)Y(t)−X(0)Y(0)−X(s)dY(s)−Y(s)dX(s).(4.57)00Theformulaforintegrationbyparts(stochasticproductrule)isgivenbyttX(t)Y(t)−X(0)Y(0)=X(s)dY(s)+Y(s)dX(s)+[X,Y](t).(4.58)00IndifferentialnotationsthisreadsdX(t)Y(t)=X(t)dY(t)+Y(t)dX(t)+d[X,Y](t).(4.59)IfdX(t)=µX(t)dt+σX(t)dB(t),(4.60)dY(t)=µY(t)dt+σY(t)dB(t),(4.61)then,asseenearlier,theirquadraticcovariationcanbeobtainedformallybymultiplicationofdXanddY,namelyd[X,Y](t)=dX(t)dY(t)=σ(t)σ(t)(dB(t))2=σ(t)σ(t)dt,XYXYleadingtotheformuladX(t)Y(t)=X(t)dY(t)+Y(t)dX(t)+σX(t)σY(t)dt.Notethatifoneoftheprocessesiscontinuousandisoffinitevariation,thenthecovariationtermiszero.Thusforsuchprocessesthestochasticproductruleisthesameasusual.Theintegrationbypartsformula(4.59)canbeestablishedrigorouslybymakingtheargumentabovemoreprecise,orbyusingItˆo’sformulaforthefunctionoftwovariablesxy,orbyapproximationsbysimpleprocesses. 114CHAPTER4.BROWNIANMOTIONCALCULUSFormula(4.57)providesanalternativerepresentationforquadraticvaria-tiont[X,X](t)=X2(t)−X2(0)−2X(s)dX(s).(4.62)0ForBrownianmotionthisformulawasestablishedinExample4.2.Itfollowsfromthedefinitionofquadraticvariation,thatitisanon-decreasingprocessint,andconsequentlyitisoffinitevariation.Itisalsoobviousfrom(4.62)thatitiscontinuous.Bythepolarizationidentity,covari-ationisalsocontinuousandisoffinitevariation.Example4.21:X(t)hasstochasticdifferentialdX(t)=B(t)dt+tdB(t),X(0)=0.WefindX(t),giveitsdistribution,itsmeanandcovariance.X(t)=tB(t)satisfiestheaboveequation,sincetheproductruleforstochasticdifferentialsisthesameasusual,whenoneoftheprocessesiscontinuousandoffinitevariation.ThusX(t)=tB(t)isGaussian,withmeanzero,andcovariancefunctionγ(t,s)=Cov(X(t),X(s))=E(X(t)X(s))=E(B(t)B(s))=Cov(B(t)B(s))=min(t,s).Example4.22:LetY(t)havestochasticdifferential1dY(t)=Y(t)dt+Y(t)dB(t),Y(0)=1.2LetX(t)=tB(t).Wefindd(X(t)Y(t)).B(t)Y(t)isaGeometricBrownianmotione(seeExample4.17).Ford(X(t)Y(t))usetheproductrule.Weneedtheexpressionford[X,Y](t).1d[X,Y](t)=dX(t)dY(t)=(B(t)dt+tdB(t))Y(t)dt+Y(t)dB(t)21212=B(t)Y(t)(dt)+B(t)Y(t)+tY(t)dB(t)dt+tY(t)(dB(t))=tY(t)dt,222as(dB(t))=dtandalltheothertermsarezero.Thusd(X(t)Y(t))=X(t)dY(t)+Y(t)dX(t)+d[X,Y](t)=X(t)dY(t)+Y(t)dX(t)+tY(t)dt,andsubstitutingtheexpressionsforXandYtheanswerisobtained.2Example4.23:LetfbeaCfunctionandB(t)Brownianmotion.Wefindquadraticcovariation[f(B),B](t).Wefindtheanswerbydoingformalcalculations.UsingItˆo’sformula1df(B(t))=f(B(t))dB(t)+f(B(t))dt,2 4.6.ITO’SFORMULAFORITˆOPROCESSESˆ115andtheconventiond[f(B),B](t)=df(B(t))dB(t),wehave21d[f(B),B](t)=df(B(t))dB(t)=f(B(t))(dB(t))+f(B(t))dB(t)dt=f(B(t))dt.22Hereweused(dB)=dt,anddBdt=0.Thust[f(B),B](t)=f(B(s))ds.0Inamoreintuitiveway,fromthedefinitionofthecovariation,takinglimitsovershrinkingpartitions
n−1nnnn[f(B),B](t)=lim(f(B(ti+1))−f(B(ti)))(B(ti+1)−B(ti))i=0
n−1f(B(tn))−f(B(tn))i+1inn2=limB(tn)−B(tn)(B(ti+1)−B(ti))i+1ii=0
n−1tnnn2≈limf(B(ti))(B(ti+1)−B(ti))=f(B(s))ds,0i=0wherewehaveusedTheorem4.14inthelastequality.Example4.24:Letf(t)beanincreasingdifferentiablefunction,andletX(t)=B(f(t)).Weshowthat[X,X](t)=[B(f),B(f)](t)=[B,B]f(t)=f(t).(4.63)Bytakinglimitsovershrinkingpartitions
n−1nn2[X,X](t)=lim(B(f(ti+1))−B(f(ti)))i=02
n−1B(f(tn))−B(f(tn))nni+1i=lim(f(ti+1)−f(ti))f(tn)−f(tn)i=0i+1i
n−1nn2=lim(f(ti+1)−f(ti))Zi=limTn,i=0B(f(tn))−B(f(tn))whereZi+1ii=areStandardNormal,andindependent,bytheprop-f(tn)−f(tn)i+1iertiesofBrownianmotion,andTn−1(f(tn)−f(tn))Z2.Thenforanyn,n=i=0i+1ii
n−1nnE(Tn)=(f(ti+1)−f(ti))=f(t).i=0 116CHAPTER4.BROWNIANMOTIONCALCULUS
n−1
n−1nnnn2Var(Tn)=Var(f(ti+1)−f(ti))=3(f(ti+1)−f(ti)),i=0i=02byindependence,andVar(Z)=3.Thelastsumconvergestozero,sincefisoffinitevariationandcontinuous,implyingthat2E(Tn−f(t))→0.2ThismeansthatthelimitinLofTnisf(t),whichimpliesthatthelimitinproba-bilityofTnisf(t),and[B(f),B(f)]=f(t).Itˆo’sFormulaforFunctionsofTwoVariablesIftwoprocessesXandYbothpossessastochasticdifferentialwithrespecttoB(t)andf(x,y)hascontinuouspartialderivativesuptoordertwo,thenf(X(t),Y(t))alsopossessesastochasticdifferential.TofinditsformconsiderformallytheTaylorexpansionofordertwo,∂f(x,y)∂f(x,y)df(x,y)=dx+dy∂x∂y)*1∂2f(x,y)∂2f(x,y)∂2f(x,y)+(dx)2+(dy)2+2dxdy.2(∂x)2(∂y)2∂x∂yNow,(dX(t))2=dX(t)dX(t)=d[X,X](t)=σ2(X(t))dt,X(dY(t))2=d[Y,Y]=σ2(Y(t))dt,anddX(t)dY(t)=d[X,Y]tYt=σX(X(t))σY(Y(t))dt,whereσX(t),andσY(t)arethediffusioncoefficientsofXandYrespectively.SowehaveTheorem4.17Letf(x,y)havecontinuouspartialderivativesuptoordertwo(aC2function)andX,YbeItˆoprocesses,then∂f∂fdf(X(t),Y(t))=(X(t),Y(t))dX(t)+(X(t),Y(t))dY(t)∂x∂y1∂2f1∂2f22+2∂x2(X(t),Y(t))σX(X(t))dt+2∂y2(X(t),Y(t))σY(Y(t))dt∂2f+(X(t),Y(t))σX(X(t))σY(Y(t))dt.(4.64)∂x∂yTheproofissimilartothatofTheorem4.13,andisomitted.Itisstressedthatdifferentialformulaehavemeaningonlythroughtheirintegralrepresentation.Example4.25:Iff(x,y)=xy,thenweobtainadifferentialofaproduct(ortheproductrule)whichgivestheintegrationbypartsformula.d(X(t)Y(t))=X(t)dY(t)+Y(t)dX(t)+σX(t)σY(t)(t)dt. 4.7.ITOPROCESSESINHIGHERDIMENSIONSˆ117AnimportantcaseofItˆo’sformulaisforfunctionsoftheformf(X(t),t).Theorem4.18Letf(x,t)betwicecontinuouslydifferentiableinx,andcon-tinuouslydifferentiableint(aC2,1function)andXbeanItˆoprocess,then∂f∂f1∂2f2df(X(t),t)=(X(t),t)dX(t)+(X(t),t)dt+σX(X(t),t)2(X(t),t)dt.∂x∂t2∂x(4.65)ThisformulacanbeobtainedfromTheorem4.17bytakingY(t)=tandobservingthatd[Y,Y]=0andd[X,Y]=0.B(t)−t/2Example4.26:WefindstochasticdifferentialofX(t)=e.x−t/2UseItˆo’sformulawithf(x,t)=e.X(t)=f(B(t),t)satisfies2∂f∂f1∂fdX(t)=df(B(t),t)=dB(t)+dt+dt∂x∂t2∂2x11=f(B(t),t)dB(t)−f(B(t),t)dt+f(B(t),t)dt22=f(B(t),t)dB(t)=X(t)dB(t).SothatdX(t)=X(t)dB(t).4.7ItˆoProcessesinHigherDimensionsdLetB(t)=(B1(t),B2(t),...,Bd(t))beBrownianmotioninIR,thatis,allcoordinatesBi(t)areindependentone-dimensionalBrownianmotions.LetFtbetheσ-fieldgeneratedbyB(s),s≤t.LetH(t)bearegularadaptedprocessd-dimensionalvectorprocess,i.e.eachofitscoordinatesissuch.Ifforeachj,T2T0Hj(t)dt<∞,thentheItˆointegrals0Hj(t)dBj(t)aredefined.Asingle2d2equivalentconditionintermsofthelengthofthevector|H|=i=1HiisT|H(t)|2dt<∞.0Itiscustomarytouseascalarproductnotation(evensuppressing·)
dT
dTH(t)·dB(t)=Hj(t)dBj(t),andH(t)·dB(t)=Hj(t)dBj(t).j=10j=10(4.66)Ifb(t)isanintegrablefunctionthentheprocess
ddX(t)=b(t)dt+Hj(t)dBj(t)j=1 118CHAPTER4.BROWNIANMOTIONCALCULUSiswelldefined.ItisascalarItˆoprocessdrivenbyad-dimensionalBrownianmotion.Moregenerally,wecanhaveanynumbernofprocessdrivenbyad-dimensionalBrownianmotion,(thevectorHi=(σi1,...σid))
ddXi(t)=bi(t)dt+σij(t)dBj(t),,i=1,...,n(4.67)j=1whereσisn×dmatrixvaluedfunction,Bisd-dimensionalBrownianmotion,X,baren-dimvector-valuedfunctions,theintegralswithrespecttoBrownianmotionareItˆointegrals.ThenXiscalledanItˆoprocess.Invectorform(4.67)becomesdX(t)=b(t)dt+σ(t)dBt.(4.68)Thedependenceofb(t)andσ(t)ontimetcanbeviathewholepathoftheprocessuptimet,pathofBs,s≤t.Theonlyrestrictionisthatthisdependenceresultsin:Tforanyi=1,2,...n,bi(t)isadaptedand0|bi(t)|dt<∞a.s.Tforanyi=1,2,...n,σ(t)isadaptedandσ2(t)dt<∞a.s.,whichassureij0ijexistenceoftherequiredintegrals.Animportantcaseiswhenthisdependenceisoftheformb(t)=b(X(t),t),σ(t)=σ(X(t),t).InthiscasethestochasticdifferentialiswrittenasdX(t)=b(X(t),t)dt+σ(X(t),t)dB(t),(4.69)andX(t)isthenadiffusionprocess,seeChapters5and6.ForItˆo’sformulaweneedthequadraticvariationofamulti-dimensionalItˆoprocesses.ItisnothardtoseethatquadraticcovariationoftwoindependentBrownianmotionsiszero.Theorem4.19LetB1(t)andB2(t)beindependentBrownianmotions.Thentheircovariationprocessexistsandisidenticallyzero.Proof:Let{tn}beapartitionof[0,t]andconsiderin
−1nnnTn=B1(ti+1)−B1(ti)B2(ti+1n)−B2(ti).i=0UsingindependenceofB1andB2,E(Tn)=0.SinceincrementsofBrownianmotionareindependent,thevarianceofthesumissumofvariances,andwehaven
−1nn2nn2Var(Tn)=EB1(ti+1)−B1(ti)EB2(ti+1)−B2(ti)i=0n
−1=(tn−tn)2≤max(tn−tn)t.i+1ii+1iii=0 4.7.ITOPROCESSESINHIGHERDIMENSIONSˆ119ThusVar(T)=E(T2)→0asδ=max(tn−tn)→0.Thisimpliesthatnnnii+1iTn→0inprobability,andtheresultisproved.
Thusfork=l,k,l=1,2,...d,[Bk,Bl](t)=0.(4.70)Using(4.70),andthebi-linearityofcovariation,itiseasytoseefrom(4.67)d[Xi,Xj](t)=dXi(t)dXj(t)=aijdt,fori,j=1,...n.(4.71)wherea,calledthediffusionmatrix,isgivenbya=σσTr,(4.72)withσTrdenotingthetransposedmatrixofσ.Itˆo’sFormulaforFunctionsofSeveralVariablesIfX(t)=(X1(t),X2(t),...,Xn(t))isavectorItˆoprocessandf(x1,x2,...,xn)isaC2functionofnvariables,thenf(X(t),X(t),...,X(t))isalsoanItˆo12nprocess,moreoveritsstochasticdifferentialisgivenbydf(X1(t),X2(t),...,Xn(t))
n∂=f(X1(t),X2(t),...,Xn(t))dXi(t)∂xii=11
n
n∂2+f(X1(t),X2(t),...,Xn(t))d[Xi,Xj](t).(4.73)2∂xi∂xji=1j=1WhenthereisonlyoneBrownianmotion,d=1,thisformulaisageneraliza-tionofItˆo’sformulaforafunctionoftwovariables(Theorem4.17).Forexamplesandapplicationsseemulti-dimensionaldiffusionsinChapters5and6.Wecommenthereontheintegrationbypartsformula.Remark4.8:(IntegrationbyParts)LetX(t)andY(t)betwoItˆoprocessesthatareadaptedtoindependentBrow-nianmotionsB1andB2.Takef(x,y)=xyandnotethatonlyoneofthe2∂xysecondderivativesisdifferentfromzero,,butthenthetermitmultiplies∂x∂yiszero,d[B1,B2](t)=0byTheorem4.19.SothecovariationofX(t)andY(t)iszero,andoneobtainsfrom(4.73)d(X(t)Y(t))=X(t)dY(t)+Y(t)dX(t),(4.74)whichistheusualintegrationbypartsformula. 120CHAPTER4.BROWNIANMOTIONCALCULUSRemark4.9:InsomeapplicationscorrelatedBrownianmotionsareused.TheseareobtainedbyalineartransformationofindependentBrownianmo-tions.IfB1andB2Wareindependent,thenthepairofprocessesB1andW=ρB1+1−ρ2B2arecorrelatedBrownianmotions.ItiseasytoseethatWisindeedaBrownianmotion,andthatd[B1,W](t)=ρdt.MoreresultsaboutItˆoprocessesinhigherdimensionsaregiveninChapter6.Remark4.10:Itˆo’sformulacanbegeneralizedtofunctionslesssmooththanC2,inparticularforf(x)=|x|.Itˆo’sformulaforf(x)=|x|becomesTanaka’sformula,andleadstotheconceptoflocaltime.Thisdevelopmentrequiresadditionalconcepts,whicharegivenlater,seeSection8.7inthegeneraltheoryforsemimartingales.Notes.MaterialinthischaptercanbefoundinGihmanandSkorohod(1972),LiptserandShiryaev(1977),(1989),KaratzasandShreve(1988),Gard(1988),RogersandWilliams(1990),(1994).4.8ExercisesExercise4.1:GivevaluesofαforwhichthefollowingprocessisdefinedtY(t)=(t−s)−αdB(s).(Thisprocessisusedinthedefinitionoftheso-called0FractionalBrownianmotion.)Exercise4.2:ShowthatifXisasimpleboundedadaptedprocess,thentX(s)dB(s)iscontinuous.0Exercise4.3:LetXnbeaGaussiansequenceconvergentindistributiontoX.ShowthatthedistributionofXiseitherNormalordegenerate.DeducethatifEX→µandVar(X)→σ2>0thenthelimitisN(µ,σ2).Sincennconvergenceinprobabilityimpliesconvergenceindistribution,deduceconver-genceofItˆointegralsofsimplenon-randomprocessestoaGaussianlimit.Exercise4.4:ShowthatifX(t)isnon-random(doesnotdependonB(t))t2tandisafunctionoftandswithX(t,s)ds<∞thenX(t,s)dB(s)isa00GaussianrandomvariableY(t).ThecollectionY(t),0≤t≤T,isaGaussianprocesswithzeromeanandcovariancefunctionforu≥0givenbytCov(Y(t),Y(t+u))=X(t,s)X(t+u,s)ds.0Exercise4.5:ShowthataGaussianmartingaleonafinitetimeinterval[0,T]isasquareintegrablemartingalewithindependentincrements.Deducet2tthatifXisnon-randomandX(s)ds<∞thenY(t)=X(s)dB(s)isa00Gaussiansquareintegrablemartingalewithindependentincrements. 4.8.EXERCISES121Exercise4.6:ObtainthealternativerelationforthequadraticvariationofItˆoprocesses,Equation(4.62),byapplyingItˆo’sformulatoX2(t).Exercise4.7:X(t)hasastochasticdifferentialwithµ(x)=bx+candσ2(x)=4x.AssumingX(t)≥0,findthestochasticdifferentialfortheprocessY(t)=X(t).Exercise4.8:AprocessX(t)on(0,1)hasastochasticdifferentialwithcoefficientσ(x)=x(1−x).Assuming00.LetY(t)=X(t)b.WhatchoiceofbwillgiveaconstantdiffusioncoefficientforY?Exercise4.10:LetX(t)=tB(t)andY(t)=eB(t).FinddX(t).Y(t)X(t)Exercise4.11:ObtainthedifferentialofaratioformuladbytakingY(t)f(x,y)=x/y.AssumethattheprocessYstaysawayfrom0.2Exercise4.12:FinddM(t),whereM(t)=eB(t)−t/2Exercise4.13:LetM(t)=B3(t)−3tB(t).ShowthatMisamartingale,firstdirectlyandthenbyusingItˆointegrals.Exercise4.14:ShowthatM(t)=et/2sin(B(t))isamartingalebyusingItˆo’sformula.Exercise4.15:Forafunctionofnvariablesandn-dimensionalBrownianmotion,writeItˆo’sformulaforf(B1(t),...,Bn(t))byusinggradientnotation∇f=(∂,...,∂).∂x1∂xnExercise4.16:Φ(x)isthestandardNormaldistributionfunction.ShowthatB(t)forafixedT>0theprocessΦ(√),0≤t≤Tisamartingale.T−ttdB(s)Exercise4.17:LetX(t)=(1−t),where0≤t<1.FinddX(t).01−sExercise4.18:LetX(t)=tB(t).Finditsquadraticvariation[X,X](t).tExercise4.19:LetX(t)=(t−s)dB(s).FinddX(t)anditsquadratic0variation[X,X](t).ComparetothequadraticvariationofItˆointegrals. Thispageintentionallyleftblank Chapter5StochasticDifferentialEquationsDifferentialequationsareusedtodescribetheevolutionofasystem.StochasticDifferentialEquations(SDEs)arisewhenarandomnoiseisintroducedintoordinarydifferentialequations(ODEs).InthischapterwedefinetwoconceptsofsolutionsofSDEs,thestrongandtheweaksolution.5.1DefinitionofStochasticDifferentialEqua-tionsOrdinaryDifferentialEquationsIfx(t)isadifferentiablefunctiondefinedfort≥0,µ(x,t)isafunctionofx,andt,andthefollowingrelationissatisfiedforallt,0≤t≤Tdx(t)=x(t)=µ(x(t),t),andx(0)=x0,(5.1)dtthenx(t)isasolutionoftheODEwiththeinitialconditionx0.Usuallytherequirementthatx(t)iscontinuousisadded.SeealsoTheorem1.4.Theaboveequationcanbewritteninotherforms.dx(t)=µ(x(t),t)dtand(bycontinuityofx(t))tx(t)=x(0)+µ(x(s),s)ds.0123 124CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONSBeforewegivearigorousdefinitionofSDEs,weshowhowtheyariseasarandomlyperturbedODEsandgiveaphysicalinterpretation.WhiteNoiseandSDEsTheWhiteNoiseprocessξ(t)isformallydefinedasthederivativeoftheBrow-nianmotion,dB(t)ξ(t)==B(t).(5.2)dtItdoesnotexistasafunctionoftintheusualsense,sinceaBrownianmotionisnowheredifferentiable.Ifσ(x,t)istheintensityofthenoiseatpointxattimet,thenitisagreedTTTthatσ(X(t),t)ξ(t)dt=σ(X(t),t)B(t)dt=σ(X(t),t)dB(t),where000theintegralisItˆointegral.StochasticDifferentialEquationsarise,forexample,whenthecoefficientsofordinaryequationsareperturbedbyWhiteNoise.Example5.1:Black-Scholes-Mertonmodelforgrowthwithuncertainrateofreturn.x(t)isthevalueof$1aftertimet,investedinasavingsaccount.Bythedefinitionofcompoundinterest,itsatisfiestheODEdx(t)/x(t)=rdt,ordx(t)/dt=rx(t),(riscalledtheinterestrate).Iftherateisuncertain,itistakentobeperturbedbynoise,r+ξ(t),andfollowingSDEisobtaineddX(t)=(r+σξ(t))X(t),dtmeaningdX(t)=rX(t)dt+σX(t)dB(t).Caseσ=0correspondstononoise,andrecoversthedeterministicequation.Thesolutionofthedeterministicequationiseasilyobtainedbyseparatingvariablesasrtx(t)=e.ThesolutiontotheaboveSDEisgivenbyageometricBrownianmotion,ascanbeverifiedbyItˆo’sformula(seeExample5.5)(r−σ2/2)t+σB(t)X(t)=e.(5.3)Example5.2:Populationgrowth.Ifx(t)denotesthepopulationdensity,thenthepopulationgrowthcanbedescribedbytheODEdx(t)/dt=ax(t)(1−x(t)).Thegrowthisexponentialwithbirthratea,whenthisdensityissmall,andslowsdownwhenthedensityincreases.RandomperturbationofthebirthrateresultsintheequationdX(t)/dt=(a+σξ(t))X(t)(1−X(t)),ortheSDEdX(t)=aX(t)(1−X(t))dt+σX(t)(1−X(t))dB(t). 5.1.DEFINITIONOFSTOCHASTICDIFFERENTIALEQUATIONS125APhysicalModelofDiffusionandSDEsThephysicalphenomenawhichgiverisetothemathematicalmodelofdiffusion(andofBrownianmotion)isthemicroscopicmotionofaparticlesuspendedinafluid.Moleculesofthefluidmovewithvariousvelocitiesandcollidewiththeparticlefromeverypossibledirectionproducingaconstantbombardment.Asaresultofthisbombardmenttheparticleexhibitsaneverpresenterraticmovement.Thismovementintensifieswithincreaseinthetemperatureofthefluid.DenotebyX(t)thedisplacementoftheparticleinonedirectionfromitsinitialpositionattimet.Ifσ(x,t)measurestheeffectoftemperatureatpointxattimet,thenthedisplacementduetobombardmentduringtime[t,t+∆]ismodelledasσ(x,t)B(t+∆)−B(t).Ifthevelocityofthefluidatpointxattimetisµ(x,t),thenthedisplacementoftheparticleduetothemovementofthefluidduringisµ(x,t)∆.ThusthetotaldisplacementfromitspositionxattimetisgivenbyX(t+∆)−x≈µ(x,t)∆+σ(x,t)B(t+∆)−B(t).(5.4)Thusweobtainfromthisequation,themeandisplacementfromxduringshorttime∆isgivenbyE(X(t+∆)−X(t))|X(t)=x≈µ(x,t)·∆,(5.5)andthesecondmomentofthedisplacementfromxduringshorttime∆isgivenby22E(X(t+∆)−X(t))|X(t)=x≈σ(x,t)∆.(5.6)Theaboverelationsshowthatforsmallintervalsoftimeboththemeanandthesecondmoment(andvariance)ofthedisplacementofadiffusingparticleattimetatpointxareproportionaltothelengthoftheinterval,withcoefficientsµ(x,t)andσ2(x,t)respectively.Itcanbeshownthat,takenasasymptoticrelationsas∆→0,thatis,replacing≈signbytheequalityandaddingtermso(∆)totheright-handsides,thesetworequirementscharacterizediffusionprocesses.Assumingthatµ(x,t)andσ(x,t)aresmoothfunctions,heuristicEquation(5.4)alsopointsoutthatforsmallintervalsoftime∆,diffusionsareapproxi-matelyGaussianprocesses.GivenX(t)=x,X(t+∆)−X(t)isapproximatelynormallydistributed,Nµ(x,t)∆,σ2(x,t)∆.Ofcourse,forlargeintervalsoftimediffusionsarenotGaussian,unlessthecoefficientsarenon-random.Astochasticdifferentialequationisobtainedheuristicallyfromtherelation(5.4)byreplacing∆bydt,and∆B=B(t+∆)−B(t)bydB(t),andX(t+∆)−X(t)bydX(t). 126CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONSStochasticDifferentialEquationsLetB(t),t≥0,beBrownianmotionprocess.AnequationoftheformdX(t)=µ(X(t),t)dt+σ(X(t),t)dB(t),(5.7)wherefunctionsµ(x,t)andσ(x,t)aregivenandX(t)istheunknownprocess,iscalledastochasticdifferentialequation(SDE)drivenbyBrownianmotion.Thefunctionsµ(x,t)andσ(x,t)arecalledthecoefficients.Definition5.1AprocessX(t)iscalledastrongsolutionoftheSDE(5.7)ifttforallt>0theintegralsµ(X(s),s)dsandσ(X(s),s)dB(s)exist,with00thesecondbeinganItˆointegral,andttX(t)=X(0)+µ(X(s),s)ds+σ(X(s),s)dB(s).(5.8)00Remark5.1:1.Astrongsolutionissomefunction(functional)F(t,(B(s),s≤t))ofthegivenBrownianmotionB(t).2.Whenσ=0,theSDEbecomesanordinarydifferentialequation(ODE).3.Anotherinterpretationof(5.7),calledtheweaksolution,isasolutionindistributionwhichwillbegivenlater.Equationsoftheform(5.7)arecalleddiffusion-typeSDEs.MoregeneralSDEshavetheformdX(t)=µ(t)dt+σ(t)dB(t),(5.9)whereµ(t)andσ(t)candependontandthewholepastoftheprocessesX(t)andB(t)(X(s),B(s),s≤t),thatis,µ(t)=µ(X(s),s≤t),t,σ(t)=σ(X(s),s≤t),t.Theonlyrestrictiononµ(t)andσ(t)isthattheymustbeadaptedprocesses,withrespectiveintegralsdefined.Althoughmanyresults(suchasexistenceanduniquenessresults)canbeformulatedforgeneralSDEs,weconcentratehereondiffusion-typeSDEs.Example5.3:WehaveseenthatX(t)=exp(B(t)−t/2)isasolutionofthestochasticexponentialSDEdX(t)=X(t)dB(t),X(0)=1.Example5.4:ConsidertheprocessX(t)satisfyingdX(t)=a(t)dB(t),whereta(t)isanon-random.Clearly,X(t)=X(0)+a(s)dB(s).Wecanrepresent0thisasafunctionoftheBrownianmotionbyintegratingbyparts,X(t)=X(0)+ta(t)B(t)−B(s)a(s)ds,assuminga(t)isdifferentiable.Inthiscasethefunction0tF(t,(x(s),s≤t))=X(0)+a(t)x(t)−x(s)a(s)ds.0ThenexttwoexamplesdemonstratehowtofindastrongsolutionbyusingItˆo’sformulaandintegrationbyparts. 5.1.DEFINITIONOFSTOCHASTICDIFFERENTIALEQUATIONS127Example5.5:(Example5.1continued)ConsidertheSDEdX(t)=µX(t)dt+σX(t)dB(t),X(0)=1.(5.10)2Takef(x)=lnx,thenf(x)=1/xandf(x)=−1/x.11122d(lnX(t))=dX(t)+−σX(t)dtX(t)2X2(t)112=µX(t)dt+σX(t)dB(t)−σdtX(t)212=(µ−σ)dt+σdB(t)2SothatY(t)=lnX(t)satisfies12dY(t)=(µ−σ)dt+σdB(t).2Itsintegralrepresentationgives12Y(t)=Y(0)+(µ−σ)t+σB(t),2and(µ−1σ2)t+σB(t)X(t)=X(0)e2.(5.11)Example5.6:(LangevinequationandOrnstein-Uhlenbeckprocess)ConsidertheSDEdX(t)=−αX(t)dt+σdB(t),(5.12)whereαandσaresomenon-negativeconstants.−αtNotethatinthecaseσ=0,thesolutiontotheODEisx0e,orinotherαtαtwordsx(t)eisaconstant.TosolvetheSDEconsidertheprocessY(t)=X(t)e.αtUsethedifferentialoftheproductrule,andnotethatthecovariationofewithαtαtX(t)iszero,asitisadifferentiablefunction(d(e)dX(t)=αedtdX(t)=0),weαtαthavedY(t)=edX(t)+αeX(t)dt.UsingtheSDEfordX(t)weobtaindY(t)=σeαtdB(t).ThisgivesY(t)=Y(0)+tσeαsdB(s).NowthesolutionforX(t)is0t−αtαsX(t)=eX(0)+σedB(s).(5.13)0TheprocessX(t)in(5.12)isknownastheOrnstein-Uhlenbeckprocess.WecanalsofindthefunctionaldependenceofthesolutionontheBrownianmotionpath.Performingintegrationbyparts,wefindthefunctiongivingthestrongsolutiont−αt−α(t−s)X(t)=F(t,(B(s),0≤s≤t))=eX(0)+σB(t)−σαeB(s)ds.0 128CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONSAmoregeneralequationisdX(t)=(β−αX(t))dt+σdB(t),(5.14)withthesolution)*tβ−αtβαsX(t)=+eX(0)−+σedB(s).(5.15)αα0UsingItˆo’sformulaitiseasytoverifythat(5.15)isindeedasolution.Example5.7:ConsidertheSDEdX(t)=B(t)dB(t).tClearly,X(t)=X(0)+B(s)dB(s),andusingintegrationbyparts(orItˆo’sfor-0mula),weobtain12X(t)=X(0)+(B(t)−t).2Remark5.2:Ifastrongsolutionexists,thenbydefinitionitisadaptedtothefiltrationofthegivenBrownianmotion,andassuchitisintuitivelyclearthatitisafunctionthepath(B(s),s≤t).ResultsofYamadaandWatanabe(1971),andKallenberg(1996)statethatprovidedtheconditionsoftheexistenceanduniquenesstheoremaresatisfied,thenthereexistsafunctionFsuchthatthestrongsolutionisgivenbyX(t)=F(t,(B(s),s≤t)).NotmuchisknownaboutFingeneral.Oftenitisnoteasytofindthisfunctiontt1/2evenforItˆointegralsX(t)=f(B(s))dB(s),e.g.X(t)=|B(s)|dB(s).00ForarepresentationofItˆointegralsasfunctionsofBrownianmotionpaths,see,forexample,RogersandWilliams(1990),p.125-127.OnlysomeclassesofSDEsadmitaclosedformsolution.Whenaclosedformsolutionishardtofind,anexistenceanduniquenessresultisimportant,becausewithoutit,itisnotclearwhatexactlytheequationmeans.Whenasolutionexistsandisunique,thennumericalmethodscanbeemployedtocomputeit.Similarlytoordinarydifferentialequations,linearSDEscanbesolvedexplicitly.5.2StochasticExponentialandLogarithmLetXhaveastochasticdifferential,andUsatisfytdU(t)=U(t)dX(t),andU(0)=1,orU(t)=1+U(s)dX(s).(5.16)0ThenUiscalledthestochasticexponentialofX,andisdenotedbyE(X).IfX(t)isoffinitevariationthenthesolutionto(5.16)isgivenbyU(t)=eX(t).ForItˆoprocessesthesolutionisgivenby 5.2.STOCHASTICEXPONENTIALANDLOGARITHM129Theorem5.2Theonlysolutionof(5.16)isgivenbyX(t)−X(0)−1[X,X](t)U(t)=E(X)(t):=e2.(5.17)Proof:Theproofofexistenceofasolutionto(5.16)consistsofverification,byusingItˆo’sformula,of(5.17).WriteU(t)=eV(t),withV(t)=X(t)−X(0)−1[X,X](t).Then2V(t)V(t)1V(t)dE(X)(t)=dU(t)=d(e)=edV(t)+ed[V,V](t).2Since[X,X](t)isoffinitevariation,andX(t)iscontinuous,[X,[X,X]](t)=0,and[V,V](t)=[X,X](t).UsingthiswiththeexpressionforV(t),weobtainV(t)1V(t)1V(t)V(t)dE(X)(t)=edX(t)−ed[X,X](t)+ed[X,X](t)=edX(t),22and(5.16)isestablished.Theproofofuniquenessisdonebyassumingthatthereisanotherprocesssatisfying(5.16),sayU1(t),andshowingbyintegrationbypartsthatd(U1(t)/U(t))=0.Itisleftasanexercise.
Notethatunlikeinthecaseoftheusualexponentialg(t)=exp(f)(t)=ef(t),thestochasticexponentialE(X)requirestheknowledgeofallthevaluesoftheprocessuptotimet,sinceitinvolvesthequadraticvariationterm[X,X](t).Example5.8:StochasticexponentialofBrownianmotionB(t)isgivenbyU(t)=B(t)−1tE(B)(t)=e2,anditsatisfiesforallt,dU(t)=U(t)dB(t)withU(0)=1.Example5.9:ApplicationinFinance:StockprocessanditsReturnprocess.LetS(t)denotethepriceofstockandassumethatitisanItˆoprocess,i.e.ithasastochasticdifferential.TheprocessofthereturnonstockR(t)isdefinedbytherelationdS(t)dR(t)=.S(t)InotherwordsdS(t)=S(t)dR(t)(5.18)andthestockpriceisthestochasticexponentialofthereturn.Returnsareusuallyeasiertomodelfromfirstprinciples.Forexample,intheBlack-Scholesmodelitisassumedthatthereturnsovernon-overlappingtimeintervalsareindependent,andhavefinitevariance.ThisassumptionleadstothemodelforthereturnprocessR(t)=µt+σB(t).ThestockpriceisthengivenbyR(t)−R(0)−1[R,R](t)S(t)=S(0)E(R)t=S0e2(µ−1σ2)t+σB(t)=S(0)e2.(5.19) 130CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONSStochasticLogarithmIfU=E(X),thentheprocessXiscalledthestochasticlogarithmofU,denotedL(U).Thisistheinverseoperationtothestochasticexponential.Forexample,thestochasticexponentialofBrownianmotionB(t)isgivenbyB(t)−1tB(t)−1te2.SoB(t)isthestochasticlogarithmofe2.Theorem5.3LetUhaveastochasticdifferentialandnottakevalue0.ThenthestochasticlogarithmofUsatisfiestheSDEdU(t)dX(t)=,X(0)=0,(5.20)U(t)moreover)*tU(t)d[U,U](t)X(t)=L(U)(t)=ln+.(5.21)U(0)02U2(t)Proof:TheSDEforthestochasticlogarithmL(U)isbythedefinitionofE(X).Thesolution(5.21)anduniquenessareobtainedbyItˆo’sformula.
B(t)Example5.10:LetU(t)=e.WefinditsstochasticlogarithmL(U)directlyB(t)1B(t)andthenverify(5.21).dU(t)=edB(t)+edt.Hence2dU(t)1dX(t)=dL(U)(t)==dB(t)+dt.U(t)2Thus1X(t)=L(U)(t)=B(t)+t.22B(t)Now,d[U,U](t)=dU(t)dU(t)=edt,sothatte2B(t)dtt11L(U)(t)=lnU(t)+=B(t)+dt=B(t)+t,2e2B(t)2200whichverifies(5.21).Remark5.3:ThestochasticLogarithmisusefulinfinancialapplications(seeKallsenandShiryaev(2002)).5.3SolutionstoLinearSDEsLinearSDEsformaclassofSDEsthatcanbesolvedexplicitly.ConsidergenerallinearSDEinonedimensiondX(t)=(α(t)+β(t)X(t))dt+(γ(t)+δ(t)X(t))dB(t),(5.22)wherefunctionsα,β,γ,δaregivenadaptedprocesses,andarecontinuousfunc-tionsoft.ExamplesconsideredintheprevioussectionareparticularcasesoflinearSDEs. 5.3.SOLUTIONSTOLINEARSDES131StochasticExponentialSDEsConsiderfindingsolutionsinthecasewhenα(t)=0andγ(t)=0.TheSDEbecomesdU(t)=β(t)U(t)dt+δ(t)U(t)dB(t).(5.23)ThisSDEisoftheformdU(t)=U(t)dY(t),(5.24)wheretheItˆoprocessY(t)isdefinedbydY(t)=β(t)dt+δ(t)dB(t).TheSDE(5.23)isthestochasticexponentialofY,seeSection(5.2).ThestochasticexponentialofYisgivenbyU(t)=E(Y)(t)1=U(0)expY(t)−Y(0)−[Y,Y](t)2ttt12=U(0)expβ(s)ds+δ(s)dB(s)−δ(s)ds0020tt12=U(0)expβ(s)−δ(s)ds+δ(s)dB(s),(5.25)020where[Y,Y](t)isobtainedfromcalculationsd[Y,Y](t)=dY(t)dY(t)=δ2(t)dt.GeneralLinearSDEsTofindasolutionfortheEquation(5.22)inthegeneralcase,lookforasolutionoftheformX(t)=U(t)V(t),(5.26)wheredU(t)=β(t)U(t)dt+δ(t)U(t)dB(t),(5.27)anddV(t)=a(t)dt+b(t)dB(t).(5.28)SetU(0)=1andV(0)=X(0).NotethatUisgivenby(5.25).Takingthedifferentialoftheproductitiseasytoseethatwecanchoosecoefficientsa(t)andb(t)insuchawaythatrelationX(t)=U(t)V(t)holds.Thedesiredcoefficientsa(t)andb(t)turnouttosatisfyequationsb(t)U(t)=γ(t),anda(t)U(t)=α(t)−δ(t)γ(t).(5.29)UsingtheexpressionforU(t),a(t)andb(t)arethendetermined.ThusV(t)isobtained,andX(t)isfoundtobe)tt*α(s)−δ(s)γ(s)γ(s)X(t)=U(t)X(0)+ds+dB(s).(5.30)0U(s)0U(s) 132CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONSLangevintypeSDELetX(t)satisfydX(t)=a(t)X(t)dt+dB(t),(5.31)wherea(t)isagivenadaptedandcontinuousprocess.Whena(t)=−αtheequationistheLangevinequation,Example5.6.WesolvetheSDEintwoways:byusingtheformula(5.30),anddirectly,similarlytoLangevin’sSDE.Clearly,β(t)=a(t),γ(t)=1,andα(t)=δ(t)=0.TofindU(t),wemustta(s)dssolvedU(t)=a(t)U(t)dt,whichgivesU(t)=e0.Thusfrom(5.30)ttua(s)ds−a(s)dsX(t)=e0X(0)+e0dB(u).0t−a(s)dsConsidertheprocesse0X(t)anduseintegrationbyparts.Thet−a(s)dsprocesse0iscontinuousandisoffinitevariation.ThereforeithaszerocovariationwithX(t),hence)*ttt−a(s)ds−a(s)ds−a(s)dsde0X(t)=e0dX(t)−a(t)e0X(t)dtt−a(s)ds=e0dB(t).Integratingweobtainttu−a(s)ds−a(s)dse0X(t)=X(0)+e0dB(u),0andfinallytttua(s)dsa(s)ds−a(s)dsX(t)=X(0)e0+e0e0dB(u).(5.32)0BrownianBridgeBrownianBridge,orpinnedBrownianmotionisasolutiontothefollowingSDEb−X(t)dX(t)=dt+dB(t),for0≤t0,t≥0hasastrongsolutionX(t)≡0.Forr≥1/2,thisistheonlystrongsolutionbyTheorem5.5.Thereforethearenoweaksolutionsotherthanzero.For00,functionsµ(x,t)andσ(x,t)areboundedandcontinuousthentheSDE(5.45)hasatleastoneweaksolutionstartingattimesatpointx,foralls,andx.Ifinadditiontheirpartialderivativeswithrespecttoxuptoordertwoarealsoboundedandcontinuous,thentheSDE(5.45)hasauniqueweaksolutionstartingattimesatpointx.MoreoverthissolutionhasthestrongMarkovproperty.TheseresultsareprovedinStroockandVaradhan(1979),ch.6.Butbetterconditionsareavailable,StroockandVaradhan(1979)Corollary6.5.5,seealsoPinsky(1995)Theorem1.10.2.Theorem5.11Ifσ(x,t)ispositiveandcontinuousandforanyT>0thereisKTsuchthatforallx∈IR|µ(x,t)|+|σ(x,t)|≤KT(1+|x|)(5.48)thenthereexistsauniqueweaksolutiontoSDE(5.45)startingatanypointx∈IRatanytimes≥0,moreoverithasthestrongMarkovproperty.CanonicalSpaceforDiffusionsSolutionstoSDEsordiffusionscanberealizedontheprobabilityspaceofcontinuousfunctions.Weindicate:howtodefineprobabilityonthisspacebymeansofatransitionfunction,howtofindthetransitionfunctionfromthegivenSDEandhowtoverifythattheconstructedprocessindeedsatisfiesthegivenSDE. 5.7.CONSTRUCTIONOFWEAKSOLUTIONS139ProbabilitySpace(Ω,F,IF)WeaksolutionscanbeconstructedonthecanonicalspaceΩ=C[0,∞)ofcontinuousfunctionsfrom[0,∞)toIR.TheBorelσ-fieldonΩistheonegeneratedbytheopensets.Opensetsinturn,aredefinedwiththehelpofametric,forexample,anopenballofradiuscenteredatωisthesetD(ω)={ω:d(ω,ω)< }.Thedistancebetweentwocontinuousfunctionsω1andω2istakenas
∞sup|ω(t)−ω(t)|10≤t≤n12d(ω1,ω2)=.2n1+sup0≤t≤n|ω1(t)−ω2(t)|n=1ConvergenceoftheelementsofΩinthismetricistheuniformconvergenceoffunctionsonboundedclosedintervals[0,T].Diffusionsonafiniteinterval[0,T]canberealizedonthespaceC([0,T])withthemetricd(ω1,ω2)=sup|ω1(t)−ω2(t)|.0≤t≤TThecanonicalprocessX(t)isdefinedbyX(t,ω)=ω(t),0≤t<∞.Itisknown(forexample,Dudley(1989)p.356)thattheBorelσ-fieldFonC[0,∞)isgivenbyσ(X(t),0≤t<∞).Thefiltrationisdefinedbytheσ-fieldsFt=σ(X(s),0≤s≤t).ProbabilityMeasureWeoutlinetheconstructionofprobabilitymeasuresfromagiventransitionfunctionP(y,t,x,s).Inparticular,thisconstructiongivestheWienermeasurethatcorrespondstotheBrownianmotionprocess.Foranyfixedx∈IRands≥0,aprobabilityP=Px,son(Ω,F)canbeconstructedbyusingproperties1.P(X(u)=x,0≤u≤s)=1.2.P(X(t2)∈B|Ft1)=P(B,t2,X(t1),t1).ThesecondpropertyassertsthatforanyBorelsetsA,B⊂IRwehavePt1,t2(A×B):=P(X(t1)∈A,X(t2)∈B)=E(P(X(t2)∈B|Ft1)I(X(t1)∈A))=E(P(B,t2,X(t1),t1)I((X(t1)∈A))=P(dy2,t2,y1,t1)Pt1(dy1),AB 140CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONSwherePt1(C)=P(X(t1)∈C).Thisextendstothen-dimensionalcylindernsets{ω∈Ω:(ω(t1),...,ω(tn))∈Jn},whereJn⊂IR,byPt1,...,tn+1(Jn+1)=P(dyn+1,tn+1,yn,tn)Pt1,...,tn(dy1×...×dyn).Jn+1TheseprobabilitiesgivethefinitedimensionaldistributionsP((ω(t1),...,ω(tn))∈Jn).ConsistencyoftheseprobabilitiesisaconsequenceoftheChapman-Kolmogorovequationforthetransitionfunction.ThusbyKolmogorov’sextensiontheoremPcanbeextendedinauniquewaytoF.ThisprobabilitymeasureP=Px,scorrespondstotheMarkovprocessstartedatxattimes,denotedearlierbyXx(t).ThusanytransitionfunctiondefinessaprobabilitysothatthecanonicalprocessisaMarkovprocess.WedescribedinparticularaconstructionoftheWienermeasure,orBrownianmotion.TransitionFunctionUnderappropriateconditionsonthecoefficientsµ(x,t)andσ(x,t),P(y,t,x,s)isdeterminedfromapartialdifferentialequation(PDE),∂u(x,s)+Lsu(x,s)=0,(5.49)∂scalledthebackwardPDE,involvingasecondorderdifferentialoperatorLs,1∂2f∂fLf(x,s)=(Lf)(x,s)=σ2(x,s)(x,s)+µ(x,s)(x,s).(5.50)ss2∂x2∂xItfollowsfromthekeypropertyofthetransitionfunction,thattf(X(t))−(Luf)(X(u))du(5.51)sisamartingaleunderPx,swithrespecttoFtfort≥s,foranytwicecontinu-ouslydifferentiablefunctionfvanishingoutsideafiniteinterval(withcompactsupport),f∈C2(IR).KSDEontheCanonicalSpaceisSatisfiedExtraconcepts(thatoflocalmartingalesandtheirintegrals)areneededtoprovetheclaimrigorously.Themainideaisasfollows.Supposethat(5.51)holdsforfunctionsf(x)=xandf(x)=x2.(Althoughtheydon’thaveacompactsupport,theycanbeapproximatedbyC2functionsonanyfiniteKinterval).Applying(5.51)tothelinearfunction,weobtainthattY(t)=X(t)−µ(X(u),u)du(5.52)s 5.7.CONSTRUCTIONOFWEAKSOLUTIONS141isamartingale.Applying(5.51)tothequadraticfunction,weobtainthattX2(t)−σ2(X(u),u)+2µ(X(u),u)X(u)du(5.53)sisamartingale.BythecharacterizationpropertyofquadraticvariationforcontinuousmartingalesY2(t)−[Y,Y](t)isamartingale,anditfollowsfromthetaboverelationsthat[Y,Y](t)=σ2(X(u),u)du.OnecandefinetheItˆoin-sttegralprocessB(t)=dY(u)/σ(X(u),u).FromthepropertiesofstochasticsintegralsitfollowsthatB(t)isacontinuouslocalmartingalewith[B,B](t)=t.ThusbyLevy’stheoremB(t)isBrownianmotion.Puttingalloftheabovetogetherandusingdifferentialnotation,therequiredSDEisobtained.Fordetails,seeforexample,RogersandWilliams(1990),p.160,alsoStroockandVaradhan(1979).WeakSolutionsandtheMartingaleProblemTakingtherelation(5.51)asprimary,StroockandVaradhandefinedaweaksolutiontotheSDEdX(t)=µ(X(t),t)dt+σ(X(t),t)dB(t)(5.54)asasolutiontotheso-calledmartingaleproblem.Definition5.12Themartingaleproblemforthecoefficients,ortheoperatorLs,isasfollows.Foreachx∈IR,ands>0,findaprobabilitymeasurePx,sonΩ,Fsuchthat1.Px,s(X(u)=x,0≤u≤s)=1,2.ForanytwicecontinuouslydifferentiablefunctionfvanishingoutsideafiniteintervalthefollowingprocessisamartingaleunderPx,swithrespecttoFttf(X(t))−(Luf)(X(u))du.(5.55)sInthecasewhenthereisexactlyonesolutiontothemartingaleproblem,itissaidthatthemartingaleproblemiswell-posed.Example5.17:BrownianmotionB(t)isasolutiontothemartingaleproblem1d2fortheLaplaceoperatorL=2dx2,thatis,foratwicecontinuouslydifferentiablefunctionfvanishingoutsideafiniteintervalt1f(B(t))−f(B(s))ds20isamartingale.SinceBrownianmotionexistsandisdeterminedbyitsdistributionuniquely,themartingaleproblemforLiswell-posed. 142CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONSRemark5.5:Notethatifafunctionvanishesoutsideafiniteinterval,thenitsderivativesalsovanishoutsidethatinterval.Thusforatwicecontinuouslydifferentiablevanishingoutsideafiniteintervalfunctionf(f∈C2),(Lf)Ksexistsiscontinuousandvanishesoutsidethatinterval.Thisassuresthattheexpectationoftheprocessin(5.55)exists.Ifonedemandsonlythatfistwicecontinuouslydifferentiablewithboundedderivatives(f∈C2),then(Lf)bsexistsbutmaynotbebounded,andexpectationin(5.55)maynotexist.Ifonetakes(f∈C2)thenoneseekssolutionstothelocalmartingaleproblem,bandanysuchsolutionmakestheprocessin(5.55)intoalocalmartingale.LocalmartingalesarecoveredinChapter7.Astherearetwodefinitionsofweaksolutionsdefinition5.8anddefinition5.12,weshowthattheyarethesame.Theorem5.13WeaksolutionsinthesenseofDefinition5.8andinthesenseofDefinition5.12areequivalent.Proof:Wealreadyindicatedtheproofinonedirection,thatifthemar-tingaleproblemhasasolution,thenthesolutionsatisfiestheSDE.TheotherdirectionisobtainedbyusingItˆo’sformula.LetX(t)beaweaksolutioninthesenseofDefinition5.8.ThenthereisaspacesupportingBrownianmotionB(t)sothatttX(t)=X(s)+µ(X(u),u)du+σ(X(u),u)dB(u),andX(s)=x.(5.56)ssissatisfiedforallt≥s.Letfbetwicecontinuouslydifferentiablewithcompactsupport.ApplyingItˆo’sformulatof(X(t)),wehavettf(X(t))=f(X(s))+(Lf)(X(u))du+f(X(u))σ(X(u),u)dB(u).uss(5.57)Thusttf(X(t))−(Lf)(X(u))du=f(X(s))+f(X(u))σ(X(u),u)dB(u).uss(5.58)Sincefanditsderivativesvanishoutsideaninterval,say[−K,K],thefunc-tionsf(x)σ(x,u)alsovanishoutsidethisinterval,foranyu.Assumingthatσ(x,u)areboundedinxonfiniteintervalswiththesameconstantforallu,ittfollowsthat|f(x)σ(x,u)|0u(x,s)=g(y)p(y,t,x,s)dyIRisbounded,satisfiesequation(5.61)andlims↑tu(x,s)=g(x),forx∈IR.Theorem5.15Supposethatσ(x,t)andµ(x,t)areboundedandcontinuousfunctionssuchthat 144CHAPTER5.STOCHASTICDIFFERENTIALEQUATIONS(A1)σ2(x,t)≥c>0,(A2)µ(x,t)andσ2(x,t)satisfyaH¨olderconditionwithrespecttoxandt,thatis,forallx,y∈IRands,t>0|µ(y,t)−µ(x,s)|+|σ2(y,t)−σ2(x,s)|≤K(|y−x|α+|t−s|α).ThenthePDE(5.61)hasafundamentalsolutionp(y,t,x,s),whichisunique,andisstrictlypositive.Ifinadditionµ(x,t)andσ(x,t)havetwopartialderivativeswithrespecttox,whichareboundedandsatisfyaH¨olderconditionwithrespecttox,thenp(y,t,x,s)asafunctioninyandt,satisfiesthePDE2∂p1∂2∂−+σ(y,t)p−µ(y,t)p=0.(5.62)∂t2∂y2∂yTheorem5.16SupposecoefficientsofLsin(5.60)satisfyconditions(A1)and(A2)ofTheorem5.15.ThenPDE(5.61)hasauniquefundamentalso-ylutionp(y,t,x,s).ThefunctionP(y,t,x,s)=p(u,t,x,s)duuniquelyde-−∞finesatransitionprobabilityfunction.Moreover,thisfunctionhasthepropertythatforanyboundedfunctionf(x,t)twicecontinuouslydifferentiableinxand2,1oncecontinuouslydifferentiableint(f∈C(IR×[0,t]))bt∂f(y,t)P(dy,t,x,s)−f(x,s)=+Luf(y,u)P(dy,u,x,s)duIRsIR∂u(5.63)forall0≤s0,andall0≤u≤t)'''''2'*'∂f(x,u)''∂f(x,u)''∂f(x,t)'kt|x|max'','',''≤cte.(6.9)∂t∂x∂x2Proof:TheproofisgiveninPinsky(1995)forboundedcoefficientsoftheSDE,butitcanbeextendedforthiscase.WegivetheproofwhenthediffusionX(t)=B(t)isBrownianmotion.LetX(t)=B(t),thenbyItˆo’sformulaMf(t)isgivenbyt∂f(B(s),s)Mf(t)=dB(s).(6.10)0∂x∂f(x,s)Bytheboundon||,fors≤t∂x)*2∂f(B(s),s)22kt|B(s)|)E≤ctEe.∂xWritingthelastexpectationasanintegralwithrespecttothedensityoftheN(0,t)distribution,itisevidentthatitisfinite,anditsintegralover[0,t]isfinite,t)*2∂f(B(s),s)Eds<∞.(6.11)0∂xBythemartingalepropertyofItˆointegrals,(6.11)impliesthattheItˆointegral(6.10)isamartingale.OnecanprovetheresultwithoutuseofItˆo’sformula,bydoingcalculationsofintegralswithrespecttoNormaldensities(seeforexample,RogersandWilliams(1987)p.36).
Corollary6.4Letf(x,t)solvethebackwardequation∂fLtf(x,t)+(x,t)=0,(6.12)∂tandconditionsofeitherofthetwotheoremsabovehold.Thenf(X(t),t)isamartingale. 152CHAPTER6.DIFFUSIONPROCESSES1Example6.1:LetX(t)=B(t),then(Lf)(x)=f(x).SolutionstoLf=0are2linearfunctionsf(x)=ax+b.Hencef(B(t))=aB(t)+bisamartingale,whichisalsoobviousfromthefactthatB(t)isamartingale.x−t/2Example6.2:LetX(t)=B(t).Thefunctionf(x,t)=esolvesthebackwardequation21∂f∂f(x,t)+(x,t)=0.(6.13)2∂x2∂tTherefore,bytheaboveCorollary6.4werecovertheexponentialmartingaleofBrow-B(t)−t/2nianmotione.Corollary6.5(Dynkin’sFormula)LetX(t)satisfy(6.1).Ifthecondi-tionsofeitheroftheabovetheoremshold,thenforanyt,0≤t≤T,t∂fEf(X(t),t)=f(X(0),0)+ELuf+(X(u),u)du.(6.14)0∂tTheresultisalsotrueiftisreplacedbyaboundedstoppingtimeτ,0≤τ≤T.Proof:Theboundsonthegrowthofthefunctionanditspartialderivativesareusedtoestablishintegrabilityoff(X(t),t)andothertermsin(6.14).SinceMf(t)isamartingale,theresultfollowsbytakingexpectations.ForboundedstoppingtimestheresultfollowsbytheOptionalStoppingTheorem,giveninChapter7.
1Example6.3:WeshowthatJ=sdB(s)hasaNormalN(0,1/3)distribution,0uJbyfindingitsmomentgeneratingfunctionm(u)=E(e).ConsidertheItˆointegraltX(t)=sdB(s),t≤1,andnoticethatJ=X1.AsdX(t)=tdB(t),X(t)isanItˆo0uxprocesswithµ(x,t)=0andσ(x,t)=t.Takef(x,t)=f(x)=e.Thisfunction122uxsatisfiesconditionsofTheorem6.3.ItiseasytoseethatLtf(x,t)=tue,note2∂fthat=0.ThereforebyDynkin’sformula∂ttuX(t)122uX(s)Ee=1+usEeds.20uX(t)Denoteh(t)=Ee,thendifferentiationwithrespecttotleadstoasimpleequation122h(t)=uth(t),withh(0)=1.2t312t312212tuByseparatingvariables,logh(t)=usds=u.Thush(t)=e23,2023t3tt3whichcorrespondstotheN(0,)distribution.ThusX(t)=sdB(s)hasN(0,)303distribution,andtheresultfollows.1Example6.4:WeprovethatB(t)dthasN(0,1/3)distribution,seealsoEx-0111ample3.6.UsingintegrationbypartsB(t)dt=B(1)−tdB(t)=dB(t)−00011tdB(t)=(1−t)dB(t),andtheresultfollows.00 6.2.CALCULATIONOFEXPECTATIONSANDPDES1536.2CalculationofExpectationsandPDEsResultsinthissectionprovideamethodforcalculationofexpectationsofafunctionorafunctionalofadiffusionprocessontheboundary.Thisex-pectationcanbecomputedbyusingasolutiontothecorrespondingpartialdifferentialequationwithagivenboundarycondition.ThisconnectionshowsthatsolutionstoPDEscanberepresentedasfunctions(functionals)ofthecorrespondingdiffusion.LetX(t)beadiffusionsatisfyingtheSDEfort>s≥0,dX(t)=µ(X(t),t)dt+σ(X(t),t)dB(t),andX(s)=x.(6.15)BackwardPDEandEg(X(T))|X(t)=xWegiveresultsonEg(X(T))|X(t)=x.Observefirstthatg(X(T))mustbeintegrable(E|g(X(T))|<∞)forthistomakesense.Ofcourse,ifgisbounded,thenthisistrue.ObservenextthatbytheMarkovpropertyofX(t),E(g(X(T))|X(t))=E(g(X(T))|Ft).Thelatterisamartingale,byTheorem2.31.ThelastingredientisItˆo’sformula,whichconnectsthistothePDE.Again,careistakenforItˆo’sformulatoproduceamartingaleterm,forwhichassumptionsonthefunctionanditsderivativesareneeded,seeTheorem6.3.Apartfromtheserequirements,theresultsareelegantandeasytoderive.Theorem6.6Letf(x,t)solvethebackwardequation,withLtgivenby(6.2),∂fLtf(x,t)+(x,t)=0,withf(x,T)=g(x).(6.16)∂tIff(x,t)satisfiestheconditionsofCorollary6.4,thenf(x,t)=E(g(X(T))|X(t)=x).(6.17)Proof:ByCorollary6.4f(X(t),t),s≤t≤T,isamartingale.ThemartingalepropertygivesE(f(X(T),T)|Ft)=f(X(t),t).Ontheboundaryf(x,T)=g(x),sothatf(X(T),T)=g(X(T)),andf(X(t),t)=E(g(X(T))|Ft).BytheMarkovpropertyofX(t),f(X(t),t)=E(g(X(T))|X(t)),andtheresultfollows.
154CHAPTER6.DIFFUSIONPROCESSESItistemptingtoshowthattheexpectationE(g(X(T))|X(t)=x)=f(x,t)satisfiesthebackwardPDE,thusestablishingexistenceofitssolutions.As-sumeforamomentthatwecanuseItˆo’sformulawithf(X(t),t),thent)*∂ff(X(t),t)=f(X(0),0)+Lsf+(X(s),s)ds+M(t),0∂swhereM(t)isamartingale.Asnoticedearlier,f(X(t),t)=E(g(X(T))|Ft)T∂fisamartingale.Itfollowsthatt(Lsf+∂s)(X(s),s)dsisamartingaleasadifferenceoftwomartingales.Sincetheintegralwithrespecttodsisafunctionoffinitevariation,andamartingaleisnot,thelattercanonlybetrueiftheintegraliszero,implyingthebackwardequation(6.16).TomakethisargumentpreciseoneneedstoestablishthevalidityofItˆo’sformula,i.e.smoothnessoftheconditionalexpectation.Thiscanbeseenbywritingtheexpectationasanintegralwithrespecttothedensityfunction,f(x,t)=E(g(X(T))|X(t)=x)=g(y)p(y,T,x,t)dy,(6.18)wherep(y,T,x,t)isthetransitionprobabilitydensity.Soxisnowinthetransitionfunctionandthesmoothnessinxfollows.ForBrownianmotion2(y−x)−√p(y,T,x,t)=√1e2T−tisdifferentiableinx(infinitelymanytimes)andT−ttheresultfollowsbydifferentiatingundertheintegral.ForotherGaussiandiffusionstheargumentissimilar.Itishardertoshowthisinthegeneralcase(seeTheorem5.16).Remark6.1:Theorem6.6showsthatanysolutionofthebackwardequationwiththeboundaryconditiongivenbyg(x)isgivenbytheintegralofgwithrespecttothetransitionprobabilitydensity,equation(6.18),affirmingthatthetransitionprobabilitydensityisafundamentalsolution(seeDefinition5.14).AresultsimilartoTheorem6.6isobtainedwhenzerointherhsoftheback-wardequationisreplacedbyaknownfunction−φ.Theorem6.7Letf(x,t)solve∂fLtf(x,t)+(x,t)=−φ(x),withf(x,T)=g(x).(6.19)∂tThenT''f(x,t)=Eg(X(T))+φ(X(s))ds'X(t)=x.(6.20)0TheproofissimilartotheaboveTheorem6.6andisleftasanexercise. 6.2.CALCULATIONOFEXPECTATIONSANDPDES155Feynman-KacformulaAresultmoregeneralthanTheorem6.6isgivenbytheFeynman-Kacformula.Theorem6.8(Feynman-KacFormula)Forgivenboundedfunctionsr(x,t)andg(x)letT'−r(X(u),u)du'C(x,t)=Eetg(X(T))'X(t)=x.(6.21)Assumethatthereisasolutionto∂f(x,t)+Ltf(x,t)=r(x,t)f(x,t),withf(x,T)=g(x),(6.22)∂tThenthesolutionisuniqueandC(x,t)isthatsolution.Proof:WegiveasketchoftheproofbyusingItˆo’sformulacoupledwithsolutionsofalinearSDE.Takeasolutionto(6.22)andapplyItˆo’sformula∂f∂fdf(X(t),t)=(X(t),t)+Ltf(X(t),t)dt+(X(t),t)σ(X(t),t)dB(t).∂t∂xThelasttermisamartingaleterm,sowriteitasdM(t).Nowuse(6.22)toobtaindf(X(t),t)=r(X(t),t)f(X(t),t)dt+dM(t).ThisisalinearSDEofLangevintypeforf(X(t),t)whereB(t)isreplacedbyM(t).IntegratingthisSDEbetweentandT,andusingT≥tasatimevariableandtastheorigin(see(5.32)inSection5.22)weobtainTTTsr(X(u),u)dur(X(u),u)dur(X(u),u)duf(X(T),T)=f(X(t),t)et+etetdM(s).tButf(X(T),T)=g(X(T)),andrearranging,weobtainTTs−r(X(u),u)dur(X(u),u)dug(X(T))et=f(X(t),t)+etdM(s).tAsthelasttermisanintegralofaboundedfunctionwithrespecttomartingale,itisitselfamartingalewithzeromean.TakingexpectationgivenX(t)=x,weobtainthatC(x,t)=f(x,t).Forotherproofsseeforexample,Friedman(1975)andPinsky(1995).
Remark6.2:Theexpressione−r(T−t)E(g(X(T))|X(t)=x)occursinFi-nanceasadiscountedexpectedpayoff,whererisaconstant.ThediscountingresultsinthetermrfintherhsofthebackwardPDE. 156CHAPTER6.DIFFUSIONPROCESSESExample6.5:Giveaprobabilisticrepresentationofthesolutionf(x,t)ofthePDE2122∂f∂f∂f2σx+µx+=rf,0≤t≤T,f(x,T)=x,(6.23)2∂x2∂x∂twhereσ,µandrarepositiveconstants.SolvethisPDEusingthesolutionofthecorrespondingstochasticdifferentialequation.TheSDEcorrespondingtoLisdX(t)=µX(t)dt+σX(t)dB(t).ItssolutionisX(t)=X(0)e(µ−σ2/2)t+σB(t).BytheFeynman-Kacformula'−r(T−t)2'−r(T−t)2f(x,t)=EeX(T)'X(t)=x=eE(X(T)|X(t)=x).(µ−σ2/2)(T−t)+σ(B(T)−B(t))UsingX(T)=X(t)e,weobtain22(2µ+σ2)(T−t)2(2µ+σ2−r)(T−t)E(X(T)|X(t)=x)=xe,givingf(x,t)=xe.Thefollowingresultshowsthatf(x,t)=Eg(X(T))|X(t)=xsatisfiesthebackwardPDE,andcanbefoundinGihmanandSkorohod(1972),Friedman(1975).(SeealsoTheorem6.5.3Friedman(1975)forC(x,t))Theorem6.9(Kolmogorov’sEquation)LetX(t)beadiffusionwithgen-eratorLt.Assumethatthecoefficientsµ(x,t)andσ(x,t)ofLtarelocallyLipschitzandsatisfythelineargrowthcondition(see(1.30)).Assumeinad-ditionthattheypossesscontinuouspartialderivativeswithrespecttoxuptoordertwo,andthattheyhaveatmostpolynomialgrowth(see(1.31)).Ifg(x)istwicecontinuouslydifferentiableandsatisfiestogetherwithitsderivativesapolynomialgrowthcondition,thenthefunctionf(x,t)=Eg(X(T))|X(t)=xsatisfies∂f(x,t)+Ltf(x,t)=0,intheregion0≤t0sss+ts+tXx(s+t)=x+µ(Xx(u))du+σ(Xx(u))dB(u).(6.26)sssssLetY(t)=Xx(s+t),andB(t)=B(s+t)−B(s),t≥0.ThenB(t)isas11Brownianmotionandfromtheaboveequation,Y(t)satisfiesfort≥0ttY(t)=x+µ(Y(v))dv+σ(Y(v))dB1(v),andY(0)=x.(6.27)00Puts=0in(6.26)toobtainttxxxxX0(t)=x+µ(X0(v))dv+σ(X0(v))dB(v),andX0(0)=x.(6.28)00ThusY(t)andXx(t)satisfythesameSDE.HenceY(t)andXx(t)havethe00samedistribution.Thereforefort>sP(y,t,x,s)=P(Xx(t)≤y)=P(Y(t−s)≤y)sx=P(X0(t−s)≤y)=P(y,t−s,x,0).
Sincethetransitionfunctionofahomogeneousdiffusiondependsontandsonlythrought−s,itisdenotedasP(t,x,y)=P(y,t+s,x,s)=P(y,t,x,0)=P(X(t)≤y|X(0)=x),(6.29)anditgivestheprobabilityfortheprocesstogofromxto(−∞,y]duringtimet.Itsdensityp(t,x,y),whenitexists,isthedensityoftheconditionaldistributionofX(t)givenX(0)=x.ThegeneratorLofatime-homogeneousdiffusionisgivenby12Lf(x)=σ(x)f(x)+µ(x)f(x).(6.30)2Underappropriateconditions(conditions(A1)and(A2)ofTheorem5.15)p(t,x,y)isthefundamentalsolutionofthebackwardequation(5.61),whichbecomesinthiscase∂p1∂2p∂p2=Lp=σ(x)+µ(x).(6.31)∂t2∂x2∂x 158CHAPTER6.DIFFUSIONPROCESSESIfmoreover,σ(x)andµ(x)havederivatives,σ(x),µ(x),andσ(x),whichareboundedandsatisfyaH¨oldercondition,thenp(t,x,y)satisfiestheforwardequationintandyforanyfixedx,whichbecomes2∂p1∂2∂(t,x,y)=σ(y)p(t,x,y)−µ(y)p(t,x,y).(6.32)∂t2∂y2∂yIntermsofthegenerator,thebackwardandtheforwardequationsarewrittenas∂p∂p∗=Lp,=Lp,(6.33)∂t∂twhere2∗1∂2∂(Lf)(y)=σ(y)f(y)−µ(y)f(y),2∂y2∂ydenotestheoperatorappearinginequation(6.32),andisknownasthead-jointoperatortoL.(Theadjointoperatorisdefinedbytherequirementthatwheneverthefollowingintegralsexist,g(x)Lf(x)dx=f(x)L∗g(x)dx.)1d2Example6.6:ThegeneratorofBrownianmotionL=2dx2iscalledtheLaplacian.Thebackwardequationforthetransitionprobabilitydensityis2∂p1∂p=Lp=.(6.34)∂t2∂x2SincethedistributionofB(t)whenB(0)=xisN(x,t),thetransitionprobabilitydensityisgivenbythedensityofN(x,t),andisthefundamentalsolutionofPDE(6.34)21(y−x)−p(t,x,y)=√e2t.2πt∗TheadjointoperatorListhesameasL,sothatLisself-adjoint.AstrongerresultthanTheorem6.9holds.Itispossibletoshow,seeforexample,KaratzasandShreve2−ay(1988)p.255,thatife|g(y)|dy<∞forsomea>0,thenf(x,t)=Exg(B(t))fort<1/2asatisfiestheheatequationwithinitialconditionf(0,x)=g(x),x∈IR.AresultofWidder1944,statesthatanynon-negativesolutiontotheheatequationcanberepresentedasp(t,x,y)dF(y)forsomenon-decreasingfunctionF.Example6.7:TheBlack-ScholesSDEdX(t)=µX(t)dt+σX(t)dB(t)forconstantsµandσ.Thegeneratorofthisdiffusionis122Lf(x)=σxf(x)+µxf(x).(6.35)2ItsdensityisthefundamentalsolutionofthePDE2∂p122∂p∂p=σx+µx.∂t2∂x2∂xThetransitionprobabilityfunctionofX(t)wasfoundinExample5.14.Itsdensityln(y/x)−(µ−σ2/2)(t−s)∂isp(t,x,y)=Φ(√).∂yσt−s 6.3.TIMEHOMOGENEOUSDIFFUSIONS159Itˆo’sFormulaandMartingalesIfX(t)isasolutionof(6.25)thenItˆo’sformulatakestheform:foranytwicecontinuouslydifferentiablef(x)df(X(t))=Lf(X(t))dt+f(X(t))σ(X(t))dB(t).(6.36)Theorem6.2andTheorem6.3fortimehomogeneousdiffusionsbecomesTheorem6.11LetXbeasolutiontoSDE(6.25)withcoefficientssatisfyingconditionsofTheorem5.4,thatis,µ(x)andσ(x)areLipschitzandsatisfythelineargrowthcondition|µ(x)|+|σ(x)|≤K(1+|x|).Iff(x)istwicecontin-uouslydifferentiableinxwithderivativesgrowingnotfasterthanexponentialsatisfyingcondition(6.9),thenthefollowingprocessisamartingale.tMf(t)=f(X(t))−Lf(X(u))du.(6.37)0Weaksolutionsto(6.25)aredefinedassolutiontothemartingaleproblem,byrequiringexistenceofafilteredprobabilityspace,withanadaptedprocessX(t),sothattf(X(t))−Lf(X(u))du(6.38)0isamartingaleforanytwicecontinuouslydifferentiablefvanishingoutsideafiniteinterval,seeSection5.8).Equation(6.38)alsoallowsustoidentifygenerators.Remark6.3:TheconceptofgeneratorisacentralconceptinstudiesofMarkovprocesses.Thegeneratorofatime-homogeneousMarkovprocess(notnecessarilyadiffusionprocess)isalinearoperatordefinedby:Ef(X(t))|X0=x−f(x)Lf(x)=lim.(6.39)t→0tIftheabovelimitexistswesaythatfisinthedomainofthegenerator.IfX(t)solves(6.25)andfisboundedandtwicecontinuouslydifferentiable,thenfrom(6.39)thegeneratorforadiffusionisobtained.Thiscanbeseenbyinterchangingthelimitandtheexpectation(dominatedconvergence),usingTaylor’sformula.Generatorsofpurejumpprocesses,suchasbirth-deathprocesses,aregivenlater.ForthetheoryofconstructionofMarkovprocessesfromtheirgeneratorsandtheirstudiesseeforexample,Dynkin(1965),EthierandKurtz(1986),StroockandVaradhan(1979),RogersandWilliams(1990).Theresultonexistenceanduniquenessofweaksolutions(Theorem5.11)be-comes 160CHAPTER6.DIFFUSIONPROCESSESTheorem6.12Ifσ(x)ispositiveandcontinuousandforanyT>0thereisKTsuchthatforallx∈IR|µ(x)|+|σ(x)|≤KT(1+|x|)(6.40)thenthereexistsauniqueweaksolutiontoSDE(6.25)startingatanypointx∈IR,moreoverthesolutionhasthestrongMarkovproperty.Thefollowingresultisspecificforone-dimensionalhomogeneousdiffusionsanddoesnotcarryovertohigherdimensions.Theorem6.13(Engelbert-Schmidt)TheSDEdX(t)=σ(X(t))dB(t)hasaweaksolutionforeveryinitialvalueX(0)ifandonlyifforallx∈IRtheconditionady=∞foralla>0−aσ2(x+y)impliesσ(x)=0.Theweaksolutionisuniqueiftheaboveconditionisequiv-alenttoσ(x)=0.Corollary6.14Ifσ(x)iscontinuous(onIR)orboundedawayfromzero,thentheaboveSDEhasauniqueweaksolution.Example6.8:BytheabovecorollaryTanaka’sSDE,Example5.15,dX(t)=sign(X(t))dB(t),X(0)=0,hasauniqueweaksolution.6.4ExitTimesfromanIntervalThemaintoolforstudyingvariouspropertiesofdiffusionsistheresultonexittimesfromaninterval.Defineτ(a,b)tobethefirsttimethediffusionexits(a,b),τ(a,b)=inf{t>0:X(t)∈/(a,b)}.SinceX(t)iscontinuous,X(τ(a,b))=aorb.ItwasshowninTheorem2.35thatτisastoppingtime,moreover,sincethefiltrationisright-continuous,{τ0:X(t)=a},withtheconventionthattheinfimumofanemptysetisinfinity.Clearly,τ=min(Ta,Tb)=Ta∧Tb.Thenextresultisinstrumentalforobtainingpropertiesofτ. 6.4.EXITTIMESFROMANINTERVAL161Theorem6.15LetX(t)beadiffusionwithacontinuousσ(x)>0on[a,b]andX(0)=x,a0on[a,b]andX(0)=x,a0,ornegativeforallxifC<0.Consequently,ifSisnotidenticallyconstant,S(x)ismonotone.Assumethatσ(x)iscontinuousandpositive,andµ(x)iscontinuous.ThenanyL-harmonicfunctioncomesfromthegeneralsolutionto(6.48),whichisgivenbyx)u*2µ(y)S(x)=exp−dydu,(6.50)σ2(y)andinvolvestwoundeterminedconstants.Example6.9:WeshowthatharmonicfunctionsforLaregivenby(6.50).Smustsolve12σ(x)S(x)+µ(x)S(x)=0.(6.51)2Thisequationleadsto(withh=S)2h/h=−2µ(x)/σ(x),2andprovidedµ(x)/σ(x)isintegrable,x2µ(y)−dyS(x)=eσ2(y).IntegratingagainwefindS(x). 6.4.EXITTIMESFROMANINTERVAL163Theorem6.17LetX(t)beadiffusionwithgeneratorLwithcontinuousσ(x)>0on[a,b].LetX(0)=x,a0.(6.56)Indeed,forxPx(Tb<∞)>Px(Tb0,andsimilarlyfory0on[a,b],and2dx2dxX(0)=x,a0.Notethatunderappropriateconditionsonthecoefficients,ifdiffusionexplodeswhenstartedatsomex0,thenitexplodeswhenstartedatanyx∈IR.Indeed,ifforanyx,y,Px(Ty<∞)>0(seeCorollary6.19),thenPy(τ∞<∞)≥Py(Tx<∞)Px(τ∞<∞)>0.Theresultbelowgivesnecessaryandsufficientconditionsforexplosions.ItisknownasFeller’stestforexplosions.Theorem6.23Supposeµ(x),σ(x)areboundedonfiniteintervals,andσ(x)>0andiscontinuous.Thenthediffusionprocessexplodesifandonlyifoneofthetwofollowingconditionsholds.Thereexistsx0suchthat)y2µ(s)*expdsx0x2µ(s)x0x0σ2(s)1.−∞exp−x0σ2(s)dsxσ2(y)dydx<∞.)y2µ(s)*expds∞x2µ(s)xx0σ2(s)2.x0exp−x0σ2(s)dsx0σ2(y)dydx<∞. 6.7.RECURRENCEANDTRANSIENCE167TheproofreliesontheanalysisofexittimesτnwiththeaidofTheorem6.16andtheFeynman-Kacformula(seeforexample,GihmanandSkorohod(1972),p.163,Pinsky(1995)p.213-214).Ifthedriftcoefficientµ(x)≡0,thenbothconditionsintheabovetheoremx0−2fail,sinceσ(y)dy→0asx→−∞,hencethefollowingresult.xCorollary6.24SDEsoftheformdX(t)=σ(X(t))dB(t)donotexplode.rExample6.12:ConsidertheSDEdX(t)=cX(t)dt+dB(t),c>0.Solutionsofrdx(t)=cx(t)dtexplodeifandonlyifr>1,seeExercise6.11.Hereσ(x)=1andrµ(x)=cx,c>0,andD=(α,β)=(0,∞).Itisclearthatthisdiffusiondriftsto+∞duetothepositivedriftforanyr>0.However,explosionoccursonlyinthecaseofr>1,thatis,Px(τD<∞)>0ifr>1,andPx(τD<∞)=0ifr≤1.Theintegralinpart2oftheabovetheoremisx2cr+1∞expydyx0r+1dx.x0exp2cxr+1r+1Usingl’Hˆopitalrule,itcanbeseenthatthefunctionundertheintegralisoforderx−rasx→∞.Since∞x−rdx<∞ifandonlyifr>1,theresultisestablishedx0(seePinsky(1995)).2rExample6.13:ConsidertheSDEdX(t)=X(t)dt+X(t)dB(t).Usingtheintegraltest,itcanbeseenthatifr<3/2,thereisnoexplosion,andifr>3/2thereisanexplosion.6.7RecurrenceandTransienceLetX(t)beadiffusiononIR.Therearevariousdefinitionsofrecurrenceandtransienceintheliterature,however,undertheimposedassumptionsonthecoefficients,theyareallequivalent.Definition6.25ApointxiscalledrecurrentfordiffusionX(t)iftheproba-bilityoftheprocesscomingbacktoxinfinitelyoftenisone,thatis,Px(X(t)=xforasequenceoft’sincreasingtoinfinity)=1.Definition6.26ApointxiscalledtransientfordiffusionX(t)ifPx(lim|X(t)|=∞)=1.t→∞Ifallpointsofadiffusionarerecurrent,thediffusionitselfiscalledrecurrent.Ifallpointsofadiffusionaretransient,thediffusionitselfiscalledtransient. 168CHAPTER6.DIFFUSIONPROCESSESTheorem6.27LetX(t)beadiffusiononIRsatisfyingassumptionsoftheex-istenceanduniquenessresultTheorem5.11,thatis,µ(x)andσ(x)areboundedonfiniteintervals,σ(x)iscontinuousandpositiveandµ(x),σ(x)satisfythelineargrowthcondition.Then1.Ifthereisonerecurrentpointthenallpointsarerecurrent.2.Iftherearenorecurrentpoints,thenthediffusionistransient.Toprovethisresulttwofundamentalpropertiesofdiffusionsareused.ThefirstisthestrongMarkovproperty,andthesecondisthestrongFellerproperty,whichstatesthatforanyboundedfunctionf(x),Exf(X(t))isacontinuousfunctioninxforanyt>0.Boththesepropertiesholdunderthestatedconditions.Italsocanbeseenthattherecurrenceisequivalenttotheproperty:foranyx,yPx(Ty<∞)=1,whereTyisthehittingtimeofy.BytheaboveTheorem,transienceisequivalenttotheproperty:foranyx,yPx(Ty<∞)<1,seeforexample,Pinsky(1995).TodecidewhetherPx(Ty<∞)<1,theformula(6.52)fortheprobabilityofexitfromoneendofanintervalintermsofthescalefunctionisused.Ifadiffusiondoesnotexplode,thenthehittingtimeofinfinityisdefinedasT∞=limb→∞Tb=∞andT−∞=lima→−∞Ta=∞.xu2µ(s)RecallthatS(x)=x0exp−x0σ2(s)dsdu,andthatby(6.52)Px(Tbx,thenS(x)−S(a)Px(Ty<∞)=limPx(Ty−∞,thenPx(Tα0.AresultsimilartoTheorem6.28holds.)*bu2µ(s)Theorem6.29LetL1=αexp−bσ2(s)dsdu.IfL1=∞thenthedif-fusionattainsthepointbbeforeα,foranyinitialpointx∈(α,b).IfL1<∞,b1yx2µ(s)y2µ(s)thenletL2=ασ2(y)αexp−bσ2(s)dsexpbσ2(s)dsdy.1.IfL2<∞thenforallx∈(α,b)thediffusionexits(α,b)infinitetime,moreoverPx(Tα<∞)>0.2.IfL2=∞theneither,theexittimeof(α,b)isinfiniteandlimt→∞X(t)=α,ortheexittimeof(α,b)isfiniteandPx(Tb0,forallt≥0)=1.Butforn=1Px(T0<∞)=1.Forn=2thescalefunctionS(x)isgivenbyS(x)=lnx,sothatforanyb>0P1(T00.Remark6.5:Thereisaclassificationofboundarypointsdependingonb1y2µ(s)theconstantsL1,L2andL3,whereL3=ασ2(y)expx0σ2(s)dsdy.Theboundaryαiscalled1.natural,ifL1=∞;2.attracting,ifL1<∞,L2=∞;3.absorbing,ifL1<∞,L2<∞,L3=∞;4.regular,ifL1<∞,L2<∞,L3<∞.Seeforexample,GihmanandSkorohod(1972),p.165.6.9StationaryDistributionsConsiderthediffusionprocessgivenbytheSDEdX(t)=µ(X(t))dt+σ(X(t))dB(t),withX(0)havingadistributionν(x)=P(X(0)≤x).Thedistributionν(x)iscalledstationaryorinvariantforthediffusionprocessX(t)ifforanytthedis-tributionofX(t)isthesameasν(x).IfP(t,x,y)denotesthetransitionproba-bilityfunctionoftheprocessX(t),thatis,P(t,x,y)=P(X(t)≤y|X(0)=x),thenthenaninvariantν(x)satisfiesν(y)=P(t,x,y)dν(x).(6.65)Tojustify(6.65)usethetotalprobabilityformulaandthefactthatthesta-tionarydistributionisthedistributionofX(t)forallt,P(X0≤y)=P(Xt≤y)=P(Xt≤y|X0=x)dν(x).(6.66)Ifthestationarydistributionhasadensity,π(x)=dν(x)/dx,thenπ(x)iscalledastationaryorinvariantdensity.Ifp(t,x,y)=∂P(t,x,y)/∂ydenotesthedensityofP(t,x,y),thenastationaryπsatisfiesπ(y)=p(t,x,y)π(x)dx.(6.67)Underappropriateconditionsonthecoefficients(µandσaretwicecontinu-ouslydifferentiablewithsecondderivativessatisfyingaH¨oldercondition)aninvariantdensityexistsifandonlyifthefollowingtwoconditionshold 6.9.STATIONARYDISTRIBUTIONS171x0x2µ(s)∞x2µ(s)1.−∞exp−x0σ2(s)dsdx=x0exp−x0σ2(s)dsdx=∞,∞1x2µ(s)2.−∞σ2(x)expx0σ2(s)dsdx<∞.Furthermore,ifaninvariantdensityistwicecontinuouslydifferentiable,thenitsatisfiestheordinarydifferentialequation2∗1∂2∂Lπ=0,thatis,σ(y)π−µ(y)π=0.(6.68)2∂y2∂yMoreover,anysolutionofthisequationwithfiniteintegraldefinesaninvari-antprobabilitydensity.Forrigorousproofseeforexample,Pinsky(1995),p.219andp.181.Tojustifyequation(6.68)heuristically,recallthatunderap-propriateconditionsthedensityofX(t)satisfiestheforward(Fokker-Plank)equation(5.62).Ifthesystemisinastationaryregime,itsdistributiondoesnotchangewithtime,whichmeansthatthederivativeofthedensitywithrespecttotiszero,resultinginequation(6.68).Equation(6.68)canbesolved,asitcanbereducedtoafirstorderdiffer-x2µ(y)entialequation(see(1.34)).Usingtheintegratingfactorexp−aσ2(y)dy,wefindthatthesolutionisgivenbyxC2µ(y)π(x)=expdy,(6.69)σ2(x)xσ2(y)0whereCisfoundfromπ(x)dx=1.Example6.15:ForBrownianmotioncondition1.aboveistrue,butcondition2.fails.Thusnostationarydistributionexists.Theforwardequationfortheinvariantdistributionis21∂p=0,2∂x2whichhasforitssolutionslinearfunctionsofxandnoneofthesehasafiniteintegral.Example6.16:TheforwardequationfortheOrnstein-Uhlenbeckprocessis2∂p∗12∂p∂=Lp=σ−α(xp),∂t2∂x2∂xandtheequationfortheinvariantdensityis212∂p∂σ−α(xp)=0.2∂x2∂xThesolutionisgivenbyxC−2αCα2π(x)=expdy=exp−x.(6.70)σ2σ2σ2σ20Thisshowsthatifαisnegative,nostationarydistributionexists,andifαispositive22thenthestationarydensityisNormalN(0,σ/(2α)).ThefactthatN(0,σ/(2α))isastationarydistributioncanbeeasilyverifieddirectlyfromrepresentation(5.13). 172CHAPTER6.DIFFUSIONPROCESSESRemark6.6:TheOrnstein-Uhlenbeckprocesshasthefollowingproperties:itisaGaussianprocesswithcontinuouspaths,itisMarkov,anditisstation-2ary,providedtheinitialdistributionisthestationarydistributionN(0,σ).2αStationaritymeansthatfinite-dimensionaldistributionsdonotchangewithshiftintime.ForGaussianprocessesstationarityisequivalenttothecovari-ancefunctiontobeafunctionof|t−s|only,i.e.Cov(X(t),X(s))=h(|t−s|)(seeExercise(6.3)).TheOrnstein-UhlenbeckprocessistheonlyprocessthatissimultaneouslyGaussian,Markovandstationary(seeforexampleBreiman(1968),p.350).InvariantMeasuresAmeasureνiscalledinvariantforX(t)ifitsatisfiestheequation∞ν(B)=P(t,x,B)dν(x)forallintervalsB.Inequation(6.65)intervalsof−∞theformB=(−∞,y]wereused.Thegeneralequationreducesto(6.65)ifC=ν(IR)<∞.Inthiscaseνcanbenormalizedtoaprobabilitydistribution.IfC=∞thisisimpossible.Densitiesofinvariantmeasures,whentheyexistandaresmoothenough,satisfyequation(6.67).Conversely,anypositivesolutionto(6.67)isadensityofaninvariantmeasure.ForBrownianmotionπ(x)=1isasolutionoftheequation(6.67).Thisisseenasfollows.Sincep(t,x,y)isthedensityoftheN(x,t)distribution,p(t,x,y)=√1exp((y−x)2/(2t)).Notethatforafixedy,asafunctionof2πtx,itisalsothedensityoftheN(y,t)distribution.Thereforeitintegratestounity,IRp(t,x,y)dx=1.Thusπ(x)=1isapositivesolutionoftheequation(6.67).Inthiscasethedensity1correspondstotheLebesguemeasure,whichisaninvariantmeasureforBrownianmotion.SinceIR1dx=∞,itcannotbenormalizedtoaprobabilitydensity.NotealsothatsincethemeanoftheN(y,t)distributionisy,wehave∞xp(t,x,y)dx=y.−∞Sothatπ(x)=xisalsoasolutionoftheequation(6.67),butitisnotapositivesolution,andthereforeisnotadensityofaninvariantmeasure.Aninterpretationoftheinvariantmeasurewhichisnotaprobabilitymea-suremaybegivenbythedensityofalargenumber(infinitenumber)ofpar-ticleswithlocationscorrespondingtotheinvariantmeasure,alldiffusingac-cordingtothediffusionequation.Thenatanytimethedensityoftheparticlesatanylocationwillbepreserved. 6.10.MULTI-DIMENSIONALSDES1736.10Multi-dimensionalSDEsWecovertheconceptsverybriefly,relyingonanalogywiththeone-dimensionalcase,butstatethedifferencesarisingduetotheincreaseindimension.LetX(t)beadiffusioninndimensions,describedbythemulti-dimensionalSDEdX(t)=b(X(t),t)dt+σ(X(t),t)dB(t),(6.71)whereσisn×dmatrixvaluedfunction,Bisd-dimensionalBrownianmotion,seesection4.7,X,baren-dimensionalvectorvaluedfunctions.Incoordinateformthisreads
ddXi(t)=bi(X(t),t)dt+σij(X(t),t)dBj(t),i=1,...,n,(6.72)j=1anditmeansthatforallt>0andi=1,...,nt
dtXi(t)=Xi(0)+bi(X(u),u)du+σij(X(u),u)dBj(u).(6.73)0j=10ThecoefficientsoftheSDEare:thevectorb(x,t)andthematrixσ(x,t).Anexistenceanduniquenessresultforstrongsolutions,undertheassump-tionoflocallyLipschitzcoefficientsholdsinthesameform,seeTheorem5.4,exceptforabsolutevaluesthatshouldbereplacedbythenorms.Thenormofthevectorisitslength,|b|=nb2.Thenormofthematrixσisi=1idefinedby|σ|2=trace(σσTr),withσTrbeingthetransposedofσ.ThenTrtrace(a)=i=1aii.Thematrixa=σσiscalledthediffusionmatrix.Theorem6.30IfthecoefficientsarelocallyLipschitzinxwithaconstantindependentoft,thatis,foreveryN,thereisaconstantKdependingonlyonTandNsuchthatforall|x|,|y|≤Nandall0≤t≤T|b(x,t)−b(y,t)|+|σ(x,t)−σ(y,t)|0,forallx∈IRandv=0i,j=1andb(x,t)isboundedonboundedsets.Thenthereexistsauniqueweaksolu-tionuptothetimeofexplosion.If,inaddition,thelineargrowthconditionissatisfied,thatis,foranyT>0thereisKTsuchthatforallx∈IR|b(x,t)|+|a(x,t)|≤KT(1+|x|),(6.79) 6.10.MULTI-DIMENSIONALSDES175thenthereexistsauniqueweaksolutiontothemartingaleproblem(6.78)start-ingatanypointx∈IRatanytimes≥0,moreoverthissolutionhasthestrongMarkovproperty.Sincetheweaksolutionisdefinedintermsofthegenerator,whichitselfde-pendsonσonlythrougha,theweaksolutionto(6.71)canbeconstructedusingasingleBrownianmotionprovidedthematrixaremainsthesame.IfasingleSDEisequivalenttoanumberofSDEs,heuristically,itmeansthatthereisasmuchrandomnessinad-dimensionalBrownianmotionasthereisinasingleBrownianmotion.ReplacementofasystemofSDEsbyasingleoneisshownindetailfortheBesselprocess.NotethattheequationσσTr=ahasmanysolutionsforσ,thematrixsquarerootisnon-unique.However,ifa(x,t)isnon-negativedefiniteforallxandt,andhasforentriestwicecontinuouslydifferentiablefunctionsofxandt,thenithasalocallyLipschitzsquarerootσ(x,t)ofthesamedimensionasa(x,t)(seeforexampleFriedman(1975)Theorem6.1.2).BesselProcessLetB(t)=(B1(t),B2(t),...,Bd(t))bethed-dimensionalBrownianmotion,d≥2.DenotebyR(t)itssquareddistancefromtheorigin,thatis,
dR(t)=B2(t).(6.80)ii=1TheSDEforR(t)isgivenby(usingd(B2(t))=2B(t)dB(t)+dt)
ddR(t)=ddt+2Bi(t)dBi(t).(6.81)i=1InthiscasewehaveoneequationdrivenbydindependentBrownianmotions.Clearly,b(x)=d,σ(x)is(1×d)matrix2(B1(t),B2(t),...,Bd(t)),sothatTrd2a(X(t))=σ(X(t))σ(X(t))=4i=1Bi(t)=4R(t)isascalar.Thusthe2generatorofX(t)isgivenbyL=dd+1(4x)d.Butthesamegeneratordx2dx2correspondstotheprocessX(t)satisfyingtheSDEbelowdrivenbyasingleBrownianmotiondX(t)=ddt+2X(t)dB(t).(6.82)ThereforethesquareddistanceprocessR(t)in(6.80)satisfiesSDE(6.82).ThisSDEwasconsideredinExample6.14.TheBesselprocessisdefinedas,thedistancefromtheorigin,Z(t)=dB2(t)=R(t).SinceR(t)hasi=1ithesamedistributionasX(t)givenby(6.82),byItˆo’sformula,Z(t)satisfiesd−1dZ(t)=dt+dB(t).(6.83)2Z(t) 176CHAPTER6.DIFFUSIONPROCESSESUsingtheone-dimensionalSDE(6.82)forR(t)wecandecideonthere-currence,transienceandattainabilityof0ofBrownianmotionindimensions2andhigher.ItfollowsfromExample6.14thatinoneandtwodimensionsBrownianmotionisrecurrent,butindimensionsthreeandhigheritistran-sient.ItwasalsoshowntherethatindimensiontwoandaboveBrownianmotionnevervisitszero.SeealsoKaratzasandShreve(1988),p.161-163.Itˆo’sFormula,Dynkin’sFormulanLetX(t)=(X1(t),...,Xn(t))beadiffusioninIRwithgeneratorL,(thegeneralcaseissimilar,butinwhatfollowstime-homogeneouscasewillbeconsidered).Letf:IRn→IRbeatwicecontinuouslydifferentiable(C2)function.ThenItˆo’sformulastatesthat
n∂f1
n
n∂2fdf(X(t))=(X(t))dXi(t)+aij(X(t))(X(t))dt.∂xi2∂xi∂xji=1i=1j=1(6.84)ItcanberegardedasaTaylor’sformulaexpansionwheredXidXj≡d[Xi,Xj]anddXidXjdXk≡0.Itˆo’sformulacanbewrittenwiththehelpofthegeneratoras
n
d∂fdf(X(t))=(Lf)(X(t))+(X(t))σij(X(t))dBj(t).(6.85)∂xii=1j=1TheanaloguesofTheorems6.3and6.15hold.Itisclearfromtheabove(6.85)thatifpartialderivativesoffarebounded,andσ(x)isbounded,thentf(X(t))−Lf(X(u))du(6.86)0isamartingale.(Withouttheassumptionoffunctionsbeingbounded,itisalocalmartingale).Theorem6.32SupposethattheassumptionsofTheorem6.31hold.LetD⊂nnIRbeaboundeddomain(anopenandsimplyconnectedset)inIR.LetX(0)=xanddenotebyτtheexittimefromD,τD=inf{t>0:X(t)∈∂D}.Thenforanytwicecontinuouslydifferentiablef,t∧τDf(X(t∧τD))−Lf(X(u))du(6.87)0isamartingale. 6.10.MULTI-DIMENSIONALSDES177ItcanbeshownthatunderconditionsoftheabovetheoremsupEx(τD)<∞.(6.88)x∈DAsacorollarythefollowingisobtainedTheorem6.33SupposethattheassumptionsofTheorem6.31hold.IffistwicecontinuouslydifferentiableinD,continuouson∂D,andsolvesLf=−φinDandf=gon∂D.(6.89)forsomeboundedfunctionsgandφ.Thenf(x),x∈D,hasrepresentationτDf(x)=Exg(X(τD))+Exφ(X(s))ds.(6.90)0Inparticularifφ≡0,solutionhasrepresentationasf(x)=Exg(X(τD)).TheproofisexactlythesameasforTheorem6.21inonedimension.Definition6.34Afunctionf(x)issaidtobeL-harmoniconDifitistwicecontinuouslydifferentiableonDandLf(x)=0forx∈D.Thefollowingresultfollowsfrom(6.86).Corollary6.35ForanyboundedL-harmonicfunctiononDwithboundedderivatives,f(X(t∧τD))isamartingale.13∂2Example6.17:Denoteby∆=2i=1∂x2thethree-dimensionalLaplacian.Thisioperatoristhegeneratorofthree-dimensionalBrownianmotionB(t),L=∆.LetD={x:|x|>r}.Thenf(x)=1/|x|isharmoniconD,thatisLf(x)=0forall3∂2√1x∈D.Toseethisperformdifferentiationandverifythat2=i=1∂xix2+x2+x21230atanypointx=0.NotethatinonedimensionallharmonicfunctionsfortheLaplacianarelinearfunctions,whereasinhigherdimensionstherearemanymore.Itiseasytoseethat1/|x|anditsderivativesareboundedonD,consequentlyifB(0)=x=0,then1/|B(t∧τD)|isamartingale.TheBackward(Kolmogorov’s)equationinhigherdimensionsisthesameasinnonedimension,withtheobviousreplacementofthestatevariablex∈IR.Wehaveseenthatsolutionstothebackwardequationcanbeexpressedbymeansofdiffusion,Theorem6.33.However,itisaformidabletasktoprovethatifX(t)isadiffusion,andg(x)isasmoothfunctiononDthenf(x)=ExgX(t))solvesthebackwardequation(seeforexample,Friedman(1975)). 178CHAPTER6.DIFFUSIONPROCESSESTheorem6.36Letg(x)beafunctionwithtwocontinuousderivativessatis-fyingapolynomialgrowthcondition,thatis,thefunctionanditsderivativesinabsolutevaluedonotexceedK(1+|x|m)forsomeconstantsK,m>0.LetX(t)satisfy(6.71).Assumethatcoefficientsb(x,t),σ(x,t)areLipschitzinxuniformlyint,satisfythelineargrowthcondition,andtheirtwoderivativessatisfyapolynomialgrowthcondition.Letf(x,t)=Exg(X(T))|X(t)=x(6.91)Thenfhascontinuousderivativesinx,whichcanbecomputedbydifferentiat-ing(6.91)undertheexpectationsign.Moreoverfhasacontinuousderivativeint,andsolvesthebackwardPDE∂fnLtf+=0,inIR×[0,T)∂tf(x,T)→g(x),ast↑T.(6.92)Thefundamentalsolutionof(6.92)givesthetransitionprobabilityfunctionofthediffusion(6.71).Remark6.7:(Diffusionsonmanifolds)ThePDEsabovecanalsobeconsideredwhenthestatevariablexbelongstonamanifold,ratherthanIR.Thefundamentalsolutionthencorrespondstothediffusiononthemanifoldandrepresentsthewayheatpropagatesonthatmanifold.Itturnsoutthatvariousgeometricpropertiesofthemanifoldcanbeobtainedfromthepropertiesofthefundamentalsolution,Molchanov(1975).Remark6.8:TheFeynman-Kacformulaholdsalsointhemulti-dimensionalcaseinthesamewayasinonedimension.If0intherighthandsideofthePDE(6.92)isreplacedbyrf,foraboundedfunctionr,thenthesolutionsatisfyingagivenboundaryconditionf(x,T)=g(x)hasarepresentation)*T−r(X(u),u)duf(x,t)=Eetg(X(T))|X(t)=x.SeeKaratzasandShreve(1988).Recurrence,TransienceandStationaryDistributionsPropertiesofrecurrenceandtransienceofmulti-dimensionaldiffusions,solu-tionsto(6.71),aredefinedsimilarlytotheone-dimensionalcase.However,inhigherdimensionsadiffusionX(t)isrecurrentifforanystartingpointnx∈IRtheprocesswillvisitaballaroundanypointy∈IRofradius,D(y),howeversmall,withprobabilityone.Px(X(t)∈D(y)forasequenceoft’sincreasingtoinfinity)=1. 6.10.MULTI-DIMENSIONALSDES179nnAdiffusionX(t)onIRistransientifforanystartingpointx∈IRtheprocesswillleaveanyball,howeverlarge,nevertoreturn.Itfollowsbyadiffusionanalysisofthesquaredlengthsofthemulti-dimensionalBrownianmotion(seeExample6.14)thatindimensionsoneandtwoBrownianmotionisrecurrent,butitistransientindimensionsthreeandhigher.Fortime-homogeneousdiffusionsunderconditionsofTheorem6.31onthecoefficients,recurrenceisequivalenttothepropertythattheprocessstartedatanypointxhitstheclosedballD¯(y)aroundanypointyinfinitetime.Undertheseconditions,thereisadichotomy,adiffusioniseithertransientorrecurrent.Invariantmeasuresaredefinedinexactlythesamewayasinonedimension.Stationarydistributionsarefiniteinvariantmeasures;theymayexistonlyifadiffusionisrecurrent.AdiffusionisrecurrentandadmitsastationarydistributionifandonlyiftheexpectedhittingtimeofD¯(y)fromxisfinite.Whenthispropertyholdsdiffusionisalsocalledergodicorpositiverecurrent.Ingeneraltherearenonecessaryandsufficientconditionsforrecurrenceandergodicityformulti-dimensionaldiffusions,howevertherearevarioustestsfortheseproperties.ThemethodofLyapunovfunctions,developedbyR.Z.Khasminskii,consistsoffindingasuitablefunctionf,suchthatLf≤0outsideaballaroundzero.Iflim|x|→∞f(x)=∞,thentheprocessistransient.Iffisultimatelydecreasing,thentheprocessisrecurrent.IfLf≤−forsome >0outsideaballaroundzero,withf(x)boundedfrombelowinthatdomain,thenadiffusionispositiverecurrent.ProofsconsistofanapplicationofItˆo’sformulacoupledwiththemartingaletheory(convergencepropertyofsupermartingales).SeefordetailsBhattacharya(1978),Hasminskii(1980),Pinsky(1995).HigherOrderRandomDifferentialEquationsSimilarlytoODEshigherorderrandomdifferentialequationshaveinterpreta-tionsasmulti-dimensionalSDEs.Forexample,asecondorderrandomdiffer-entialequationoftheformx¨+h(x,x˙)=B,˙(6.93)where˙x(t)=dx(t)/dt,¨x(t)=d2x(t)/dt2,andB˙denotestheWhitenoise,hasinterpretationasthefollowingtwo-dimensionalSDEbylettingx1(t)=x(t),(6.94)x2(t)=dx1(t)/dt.(6.95)dX1(t)=X2(t)dt,(6.96)dX2(t)=−h(X1(t),X2(t))+dB(t).(6.97) 180CHAPTER6.DIFFUSIONPROCESSESSuchequationsareconsideredinSection14.2ofChapter14.Highern-thorderrandomequationsareinterpretedinasimilarway:bylet-tingX1(t)=X(t)anddXi(t)=Xi+1(t)dt,i=1,...,n−1,ann-dimensionalSDEisobtained.Notes.MostofthematerialcanbefoundinFriedman(1975),GihmanandSkorohod(1982),StroockandVaradhan(1979),KaratzasandShreve(1988),RogersandWilliams(1990),Pinsky(1995).6.11ExercisesExercise6.1:Showthatforanyu,f(x,t)=exp(ux−u2t/2)solvesthebackwardequationforBrownianmotion.Takederivatives,first,second,etc.,ofexp(ux−u2t/2)withrespecttou,andsetu=0,toobtainthatfunctionsx,x2−t,x3−3tx,x4−6tx2+3t2,etc.alsosolvethebackwardequation(6.13).DeducethatB2(t)−t,B(t)3−3tB(t),B4(t)−6tB2(t)+3t2aremartingales.Exercise6.2:FindthegeneratorfortheOrnstein-Uhlenbeckprocess,writethebackwardequationandgiveitsfundamentalsolution.Verifythatitsatis-fiestheforwardequation.Exercise6.3:LetX(t)beastationaryprocess.Showthatthecovariancefunctionγ(s,t)=Cov(X(s),X(t))isafunctionof|t−s|only.Hint:takek=2.DeducethatforGaussianprocessesstationarityisequivalenttotherequirementsthatthemeanfunctionisaconstantandthecovariancefunctionisafunctionof|t−s|.Exercise6.4:X(t)isadiffusionwithcoefficientsµ(x)=cxandσ(x)=1.tGiveitsgeneratorandshowthatX2(t)−2cX2(s)ds−tisamartingale.0Exercise6.5:X(t)isadiffusionwithµ(x)=2xandσ2(x)=4x.GiveitsgeneratorL.SolveLf=0,andgiveamartingaleMf.FindtheSDEfortheprocessY(t)=X(t),andgivethegeneratorofY(t).Exercise6.6:Findf(x)suchthatf(B(t)+t)isamartingale.Exercise6.7:X(t)isadiffusionwithcoefficientsµ(x,t),σ(x,t).Findadifferentialequationforf(x,t)suchthatY(t)=f(X(t),t)hasinfinitesimaldiffusioncoefficientequalto1.Exercise6.8:Showthatthemeanexittimeofadiffusionfromaninterval,which(byTheorem6.16)satisfiestheODE(6.44)isgivenbyxybyxG(s)dsdsdsav(x)=−2G(y)dy+2G(y)dy,σ2(s)G(s)σ2(s)G(s)bG(s)dsaaaaa(6.98) 6.11.EXERCISES181x2µ(s)whereG(x)=exp−aσ2(s)ds.Exercise6.9:FindPx(Tb0,x(0)=x>0,explodesifandonlyifr>1.0Exercise6.12:InvestigateforexplosionsthefollowingprocessdX(t)=X2(t)dt+σXα(t)dB(t).Exercise6.13:ShowthatBrownianmotionB(t)isrecurrent.ShowthatB(t)+tistransient.Exercise6.14:ShowthattheOrnstein-Uhlenbeckprocessispositivelyrecur-rent.ShowthatthelimitingdistributionfortheOrnstein-Uhlenbeckprocess(5.6)exists,andisgivenbyitsstationarydistribution.Hint:thedistributiontofσe−αteαsdBisNormal,finditsmeanandvariance,andtakelimits.0sExercise6.15:ShowthatthesquareoftheBesselprocessX(t)in(6.64)comesarbitrarilyclosetozerowhenn=2,thatis,P(Ty<∞)=1foranysmally>0,butwhenn≥3,P(Ty<∞)<1.Exercise6.16:LetdiffusionX(t)haveσ(x)=1,µ(x)=−1forx<0,µ(x)=1forx>0andµ(0)=0.Showthatπ(x)=e−|x|isastationarydistributionforX.Exercise6.17:Letdiffusionon(α,β)besuchthatthetransitionprobabilitydensityp(t,x,y)issymmetricinxandy,p(t,x,y)=p(t,y,x)forallx,yandt.Showthatif(α,β)isafiniteinterval,thentheuniformdistributionisinvariantfortheprocessX(t).Exercise6.18:Investigateforabsorptionatzerothefollowingprocess(usedasamodelforinterstrates,thesquarerootmodelofCox,IngersollandRoss).dX(t)=b(a−X(t))dt+σX(t)dB(t),whereparametersb,aandσarecon-stants. Thispageintentionallyleftblank Chapter7MartingalesMartingalesplayacentralroleinthemoderntheoryofstochasticprocessesandstochasticcalculus.MartingalesconstructedfromaBrownianmotionwereconsideredinSection3.3andmartingalesarisingindiffusionsinSection6.1.Martingaleshaveaconstantexpectation,whichremainsthesameunderrandomstopping.Martingalesconvergealmostsurely.Stochasticintegralsaremartingales.Thesearethemostimportantpropertiesofmartingales,whichholdundersomeconditions.7.1DefinitionsThemainingredientinthedefinitionofamartingaleistheconceptofcondi-tionalexpectation,consultChapter2foritsdefinitionandproperties.Definition7.1AstochasticprocessM(t),wheretimetiscontinuous0≤t≤T,ordiscretet=0,1,...,T,adaptedtoafiltrationIF=(Ft)isamartingaleifforanyt,M(t)isintegrable,thatis,E|M(t)|<∞andforanytandswith0≤sn)=0.(7.2)n→∞Indeed,ifXisintegrablethen(7.2)holdsbythedominatedconvergence,sincelimn→∞|X|I(|X|>n)=0and|X|I(|X|>n)≤|X|.Conversely,letnbelargeenoughfortherhsin(7.2)tobefinite.ThenE|X|=E|X|I(|X|>n)+E|X|I(|X|≤n)<∞,sincethefirsttermisfiniteby(7.2)andthesecondisboundedbyn.Definition7.5AprocessX(t),0≤t≤TiscalleduniformlyintegrableifE|X(t)|I(|X(t)|>n)convergestozeroasn→∞uniformlyint,thatis,limsupE|X(t)|I(|X(t)|>n)=0,(7.3)n→∞twherethesupremumisover[0,T]inthecaseofafinitetimeintervaland[0,∞)iftheprocessisconsideredon0≤t<∞.Example7.2:WeshowthatifX(t),0≤t≤Tisuniformlyintegrable,thenitisintegrable,thatis,suptE|X(t)|<∞.Indeed,supE|X(t)|n)+n.ttSinceX(t)isuniformlyintegrable,thefirsttermconvergestozeroasn→∞,inparticularitisbounded,andtheresultfollows.Sufficientconditionsforuniformintegrabilityaregivennext.Theorem7.6IftheprocessXisdominatedbyanintegrablerandomvariable,|X(t)|≤YandE(Y)<∞,thenitisuniformlyintegrable.Inparticular,ifE(supt|X(t)|)<∞,thenitisuniformlyintegrable.Proof:E|X(t)|I(|X(t)|>n)n)→0,asn→∞.
Notethatthereareuniformlyintegrableprocesses(martingales)whicharenotdominatedbyanintegrablerandomvariable,sothatthesufficientconditionforuniformintegrabilityE(supt|X(t)|)<∞isnotnecessaryforuniformin-tegrability.Anothersufficientconditionforuniformintegrabilityisgivenbythefollowingresult,seeforexampleProtter(1992),p.9,LiptserandShiryaev(2001),p.17. 186CHAPTER7.MARTINGALESTheorem7.7Ifforsomepositive,increasing,convexfunctionG(x)on[0,∞)suchthatlimx→∞G(x)/x=∞,supEG(|X(t)|)<∞,(7.4)t≤TthenX(t),t≤Tisuniformlyintegrable.Weomittheproof.InpracticetheaboveresultisusedwithG(x)=xrforr>1,anduniformintegrabilityischeckedbyusingmoments.Forsecondmomentsr=2,wehave:squareintegrabilityimpliesuniformintegrability.Corollary7.8IfX(t)issquareintegrable,thatis,supEX2(t)<∞,thenittisuniformlyintegrable.Inviewofthis,examplesofuniformlyintegrablemartingalesareprovidedbysquareintegrablemartingalesgiveninExample7.1.Thefollowingresultprovidesaconstructionofuniformlyintegrablemartingales.Theorem7.9(Doob’s,Levy’smartingale)LetYbeanintegrableran-domvariable,thatis,E|Y|<∞anddefineM(t)=E(Y|Ft).(7.5)ThenM(t)isauniformlyintegrablemartingale.Proof:ItiseasytoseethatM(t)isamartingale.Indeed,bythelawofdoubleexpectation,E(M(t)|Fs)=EE(Y|Ft)|Fs=E(Y|Fs)=M(s).Theproofofuniformintegrabilityismoreinvolved.Itisenoughtoes-tablishtheresultforY≥0asthegeneralcasewillfollowbyconsider-ingY+andY−.IfY≥0thenM(t)≥0forallt.WeshownextthatM∗=supM(t)<∞.Ifnot,thereisasequenceoft↑∞suchthatt≤TnM(tn)↑∞.Bymonotoneconvergence,EM(tn)↑∞,whichisacontradic-tion,asEM(tn)=EY<∞.Now,bythegeneraldefinitionofconditionalexpectation,see(2.16),EM(t)I(M(t)>n)=EYI(M(t)>n).Since{M(t)>n}⊆{M∗>n},EYI(M(t)>n)≤EYI(M∗>n).ThusEM(t)I(M(t)>n)≤EYI(M∗>n).Sincetheright-handsidedoesnotdependont,supEM(t)I(M(t)>n)≤EYI(M∗>n).Butthist≤Tconvergestozeroasn→∞,becauseM∗isfiniteandYisintegrable.
Themartingalein(7.5)issaidtobeclosedbyY.AnimmediatecorollaryisCorollary7.10AnymartingaleM(t)onafinitetimeinterval0≤t≤T<∞isuniformlyintegrableandisclosedbyM(T).Itwillbeseeninthenextsectionthatauniformlyintegrablemartingaleon[0,∞)isalsooftheform(7.5).Thatis,thereexistsarandomvariable,calledM(∞)suchthatthemartingalepropertyholdsforall0≤s0andisamartingale.00SincesupE(M2(t))=suptf2(s)ds=∞f2(s)ds<∞,M(t)isuniformlyt>0t>0001integrable.ThusitconvergesalmostsurelytoY.ConvergenceisalsoinL,that∞isE|M(t)−Y|→0ast→∞.DenoteY=M(∞)=f(s)dB(s),thenwehave0∞1shownthatY−M(t)=f(s)dB(s)convergestozeroalmostsurelyandinL.Ytistheclosingvariable.Indeed,∞tE(Y|Ft)=E(M(∞)|Ft)=E(f(s)dB(s)|Ft)=f(s)dB(s)=M(t).00Example7.5:AboundedpositivemartingaleM(t)=E(I(Y>0)|Ft)with∞∞2Y=f(s)dB(s),wheref(s)isnon-randomandf(s)ds<∞,fromthe00previousexample.M(t)=E(I(Y>0)|Ft)=P(Y>0|Ft))∞t'*'=Pf(s)dB(s)>−f(s)dB(s)'Ftt0tf(s)dB(s)=Φ,0,(7.6)∞f2(s)dstwherethelastequalityisduetonormalityoftheItˆointegralforanon-randomf.Bytakingftobezeroon(T,∞),aresultisobtainedformartingalesoftheformTE(I(f(s)dB(s)>0)|Ft).Inparticular,bytakingf(s)=1[0,T](s),weobtainthat0√Φ(B(t)/T−t)isapositiveboundedmartingaleon[0,T].Itsdistributionfort≤Tisleftasanexercise. 7.4.OPTIONALSTOPPING1897.4OptionalStoppingInthissectionweconsiderresultsonstoppingmartingalesatrandomtimes.Recallthatarandomtimeτiscalledastoppingtimeifforanyt>0thesets{τ≤t}∈Ft.ForfiltrationsgeneratedbyaprocessX,τisastoppingtimeifitispossibletodecidewhetherτhasoccurredornotbyobservingtheprocessuptotimet.AmartingalestoppedatarandomtimeτistheprocessM(t∧τ).ABasicStoppingresult,givenherewithoutproof,statesthatamartingalestoppedatastoppingtimeisamartingale,inparticularEM(τ∧t)=EM(0).Thisequationisusedmostfrequently.Theorem7.14IfM(t)isamartingaleandτisastoppingtime,thenthestoppedprocessM(τ∧t)isamartingale.Moreover,EM(τ∧t)=EM(0).(7.7)Thisresultwasprovedindiscretetime(seeTheorem3.39).Wereferto(7.7)astheBasicStoppingequation.Remark7.1:WestressthatinthistheoremM(τ∧t)isamartingalewithrespecttotheoriginalfiltrationFt.SinceitisadaptedtoFτ∧t,itisalsoanFτ∧t-martingale(seeExercise7.1).Example7.6:(ExitofBrownianMotionfromanInterval)LetB(t)beBrownianmotionstartedatxandτbethefirsttimewhenB(t)exitstheinterval(a,b),aKandapplying(7.7),EM(0)=EM(t∧τ)=EM(τ).Appliedtogamblingthisshowsthatwhenbettingonamartingale,onaveragenolossorgainismade,evenifacleverstoppingruleisused,provideditisbounded.Wedon’tgiveaproofofthesecondstatement,thedifficultpointisinshowingthatM(τ)isintegrable.Theorem7.16LetM(t)beamartingaleandτafinitestoppingtime.IfE|M(τ)|<∞,andlimEM(t)I(τ>t)=0,(7.8)t→∞thenEM(τ)=EM(0).Proof:WriteM(τ∧t)asM(τ∧t)=M(t)I(t<τ)+M(τ)I(t≥τ).(7.9)UsingBasicStoppingResult(7.7),EM(τ∧t)=EM(0).Takingexpectationsin(7.9),wehaveEM(0)=EM(t)I(t<τ)+EM(τ)I(t≥τ).(7.10)Nowtakethelimitin(7.10)ast→∞.Sinceτisfinite,I(t≥τ)→I(τ<∞)=1.|M(τ)|I(t≥τ)≤|M(τ)|,integrable.HenceEM(τ)I(t≥τ)→EM(τ)bydominatedconvergence.ItisassumedthatEM(t)I(t<τ)→0ast→∞,andtheresultfollows.
TheBasicStoppingresultorOptionalStoppingareusedtofindthedistribu-tionofstoppingtimesforBrownianmotionandRandomWalks.Example7.8:(HittingtimesofBrownianMotion)WederivetheLaplacetransformofhittingtimes,fromwhichitalsofollowsthattheyarefinite.LetB(t)beaBrownianmotionstartingat0,andTb=inf{t:B(t)=b},uB(t)−u2t/2b>0.ConsidertheexponentialmartingaleofBrownianmotione,u>0,uB(t∧T2stoppedatTb)−(t∧Tb)u/2.UsingtheBasicStoppingresult(7.7)b,eEeuB(t∧Tb)−(t∧Tb)u2/2=1.ubThemartingaleisboundedfromabovebyeanditispositive.IfwetakeitasalreadyproventhatTbisfinite,P(Tb<∞)=1,thenweobtainbytakingt→∞2√ub−(t∧Tb)u/2thatEe=1.Replacinguby2u,weobtaintheLaplacetransformofTb√−uTb−b2uψT(u)=Ee=e.(7.11)b 7.4.OPTIONALSTOPPING191WenowshowthefinitenessofTb.Writetheexpectationofthestoppedmartingaleub−T22bu/2uB(t)−tu/2EeI(Tb≤t)+EeI(Tb>t)=1.(7.12)uB(t)−tu2/2ub−tu2/2ub−tu2/2ThetermEeI(Tb>t)≤EeI(Tb>t)≤e→0,ast→∞.Thustakinglimitsintheaboveequation(7.12),eub−Tbu2/2I(Tub−Tbu2/2Eb≤t)→EeI(Tb<∞)=1.Therefore−T22ebu/2I(T−ub−TbuEb<∞)=e.ButeI(Tb=∞)=0,thereforebyadding−T2thisterm,wecanwriteEebu/2=e−ub.ItfollowsinparticularthatP(Tb<∞)=limu↓0ψ(u)=1,andTbisfinite.Hence(7.11)isproved.ThedistributionofTbcorrespondingtothetransform(7.11)isgiveninTheorem3.18.ThefollowingresultisinsomesensetheconversetotheOptionalStoppingTheorem.Theorem7.17LetX(t),t≥0,besuchthatforanyboundedstoppingtimeτ,X(τ)isintegrableandEX(τ)=EX(0).ThenX(t),t≥0isamartingale.Proof:Theproofconsistsofcheckingthemartingalepropertybyusingappropriatestoppingtimes.Sinceadeterministictimetisastoppingtime,X(t)isintegrable.WithoutlossofgeneralitytakeX(0)=0.Nextweshowthatfort>s,E(X(t)|Fs)=X(s).Inotherwords,weneedtoshowthatforanyss,defineastoppingtimeτ=sI(B)+tI(Bc).sWehaveE(X(τ))=EX(s)I(B)+EX(t)I(Bc).SinceEX(τ)=0,EX(s)I(B)=EX(τ)−EX(t)I(Bc)=−EX(t)I(Bc).Astherighthandsideoftheaboveequalitydoesnotdependons,itfollowsthat(7.13)holds.
ThefollowingresultissometimesknownastheOptionalSamplingTheo-rem(seeforexample,RogersandWilliams(1990)).Theorem7.18(OptionalSampling)LetM(t)beauniformlyintegrablemartingale,andτ1≤τ2≤∞twostoppingtimes.ThenEM(τ2)|Fτ1=M(τ1),a.s.(7.14)OptionalStoppingofDiscreteTimeMartingalesWeconsidernextthecaseofdiscretetimet=0,1,2...,andmartingalesarisinginaRandomWalk. 192CHAPTER7.MARTINGALESGambler’sRuinConsideragameplayedbytwopeoplebybettingontheoutcomesoftossesofacoin.Youwin$1ifHeadscomeupandlose$1ifTailscomeup.Thegamestopswhenonepartyhasnomoneyleft.Youstartwithx,andyouropponentwithbdollars.ThenSn,theamountofmoneyyouhaveattimenisaRandomWalk(seeSection3.12).TheGambler’sruinproblemistofindtheprobabilitiesofruinoftheplayers.Inthisgamethelossofonepersonisthegainoftheother(azerosumgame).Assumingthatthegamewillendinafinitetimeτ(thisfactwillbeshownlater),itfollowsthattheruinprobabilitiesoftheplayersadduptoone.Considerfirstthecaseofthefaircoin.Then
n11Sn=x+ξi,P(ξi=1)=,P(ξi=−1)=,22i=1isamartingale(seeTheorem3.33).Letτbethetimewhenthegamestops,thefirsttimetheamountofmoneyyouhaveisequalto0(yourruin)orx+b(youropponent’sruin).Thenτisastoppingtime.Denotebyutheprobabilityofyourruin.Itistheprobabilityofyoulosingyourinitialcapitalxbeforewinningbdollars.ThusP(Sτ=0)=uandP(Sτ=x+b)=1−u.(7.15)FormallyapplyingtheOptionalStoppingTheoremE(Sτ)=S0=x.(7.16)ButE(Sτ)=(x+b)×(1−u)+0×u=(x+b)u.Theseequationsgivebu=.(7.17)x+bSothattheruinprobabilitiesaregivenbyasimplecalculationusingmartingalestopping.Wenowjustifythesteps.Snisamartingale,andτisastoppingtime.ByTheorem7.14thestoppedprocessSn∧τisamartingale.Itisnon-negativeandboundedbyx+b,bythedefinitionofτ.ThusSn∧τisauniformlyintegrablemartingale.HenceitconvergesalmostsurelytoafinitelimitY,(withEY=x),limS=Y.ByTheorem3.33S2−nisamartingale,n→∞n∧τnandsoisS2−n∧τ.Thusforallnbytakingexpectationn∧τE(S2)=E(n∧τ)+E(S2).(7.18)n∧τ0 7.4.OPTIONALSTOPPING193Bydominatedconvergence,thelhshasafinitelimit,thereforethereisafinitelimitlimn→∞E(n∧τ).Expandingthis,E(n∧τ)≥nP(τ>n),wecanseethatforalimittoexistitmustbelimP(τ>n)=0,(7.19)n→∞sothatP(τ<∞)=1,andτisfinite.(NotethatastandardproofoffinitenessofτisdonebyusingMarkovChainTheory,thepropertyofrecurrenceofstatesinaRandomWalk.)WritingE(Sn∧τ)=xandtakinglimitsasn→∞,theequation(7.16)isobtained(alternatively,theconditionsoftheOptionalStoppingTheoremhold).ThisconcludesarigorousderivationoftheruinprobabilityinanunbiasedRandomWalk.WenowconsiderthecasewhentheRandomWalkisbiased,p=q.
nSn=x+ξi,P(ξi=1)=p,P(ξi=−1)=q=1−p.i=1InthiscasetheexponentialmartingaleoftheRandomWalkM=(q/p)Snnisused(seeTheorem3.33).Stoppingthismartingale,weobtaintheruinprobability(q/p)b+x−(q/p)xu=.(7.20)(q/p)b+x−1JustificationoftheequationE(Mτ)=M0issimilartothepreviouscase.HittingTimesinRandomWalksLetSndenoteaRandomWalkontheintegersstartedatS0=x,Sn=nS0+i=1ξi,P(ξi=1)=p,P(ξi=−1)=q=1−p,witharbitraryp,andTbthefirsthittingtimeofb,Tb=inf{n:Sn=b}(infimumofanemptysetisinfinity).Withoutlossofgeneralitytakethestartingstatex=0,otherwiseconsidertheprocessSn−x.Considerhittingthelevelb>0,forb<0considertheprocess−Sn.WefindtheLaplacetransformofT,ψ(λ)=E(e−λTb),λ>0,bystoppingbtheexponentialmartingaleoftheRandomWalkM=euSn−nh(u),wherenh(u)=lnE(euξ1),anduisarbitrary(seeSection3.12).uSn∧T−(n∧Tb)h(u)E(Mn∧T)=Eeb=1.(7.21)bTakeu,sothath(u)=λ>0.Writetheexpectationin(7.21)asEeuSn∧Tb−(n∧Tb)h(u)=EeuSTb−Tbh(u)I(T≤n)+EeuSn−nh(u)I(T>n).bb 194CHAPTER7.MARTINGALESThefirsttermequalstoEeub−Tbh(u)I(T≤n).Thesecondtermconvergesbtozero,becausebydefinitionofTb,EeuSn−nh(u)I(T>n)≤Eeub−nh(u)I(T>n)≤eub−nh(u)→0.bbNowtakinglimitsin(7.21),usingdominatedconvergenceweobtainEe−h(u)TbI(T<∞)=e−ub.bNotethate−h(u)TbI(T=∞)=0,thereforebyaddingthisterm,wecanwritebEe−h(u)Tb=e−ub.(7.22)ThisispracticallytheLaplacetransformofTb,itremainstoreplaceh(u)byλ,bytakingu=h(−1)(λ),withh(−1)beingtheinverseofh.ThustheLaplacetransformofTbisgivenby(−1)ψ(λ)=Ee−λTb=e−h(λ)b.(7.23)Tofindh(−1)(λ),solveh(u)=λ,whichisequivalenttoE(euξ1)=eλ,orpeu+(1−p)e−u=eλ.Therearetwovaluesforeu=(eλ±e2λ−4p(1−p))/(2p),butonlyonecorrespondstoaLaplacetransform,(7.22).Thuswehaveb2pψ(λ)=Ee−λTb=.(7.24)eλ+e2λ−4p(1−p)UsingageneralresultonLaplacetransformofarandomvariable,2pbP(Tb<∞)=limψ(λ)=.(7.25)λ↓01+|1−2p|ItnowfollowsthatthehittingtimeTbofbisfiniteifandonlyifp≥1/2.Forp<1/2,thereisapositiveprobabilitythatlevelbisneverreached,P(T=∞)=1−(p)b.b1−pWhenthehittingtimeoflevelbisfinite,itmayormaynothaveafiniteexpectation.Ifp≥1/2wehave%lE(T)=−ψ(0)=2p−1ifp>1/2b∞ifp=1/2.Thuswehaveshownthatwhenp≥1/2anypositivestatewillbereachedfrom0inafinitetime,butwhenp=1/2theaveragetimeforittohappenisinfinite.Theresultsobtainedaboveareknownastransience(p=1/2)andrecur-rence(p=1/2)oftheRandomWalk,andareusuallyobtainedbyMarkovChainsTheory. 7.5.LOCALIZATIONANDLOCALMARTINGALES195Example7.9:(Optionalstoppingofdiscretetimemartingales)LetM(t)beadiscretetimemartingaleandτbeastoppingtimesuchthatE|M(τ)|<∞.1.IfEτ<∞and|M(t+1)−M(t)|≤K,thenEM(τ)=EM(0).2.IfEτ<∞andE|M(t+1)−M(t)||Ft≤K,thenEM(τ)=EM(0).Proof:Weprovethefirststatement.t−1M(t)=M(0)+M(i+1)−M(i).Thistogetherwiththeboundonincrementsi=0gives
t−1M(t)≤|M(0)|+|M(i+1)−M(i)|≤|M(0)|+Kt.i=0Takeforsimplicitynon-randomM(0)ThenEM(t)I(τ>t)≤|M(0)|P(τ>t)+KtP(τ>t).Thelasttermconvergestozero,tP(τ>t)≤E(τI(τ>t))→0,bydominatedconvergenceduetoE(τ)<∞.Thuscondition(7.8)holds,andtheresultfollows.Theproofofthesecondstatementissimilarandisleftasanexercise.
7.5LocalizationandLocalMartingalestAsitwasseenearlierinChapter4,ItˆointegralsX(s)dB(s)aremartingales0t2undertheadditionalconditionX(s)ds<∞.Ingeneral,stochasticinte-0gralswithrespecttomartingalesareonlylocalmartingalesratherthantruemartingales.Thisisthemainreasonforintroducinglocalmartingales.Wehavealsoseenthatforthecalculationofexpectationsstoppingandtruncationsareoftenused.TheseideasgiverisetothefollowingDefinition7.19ApropertyofastochasticprocessX(t)issaidtoholdlocallyifthereexistsasequenceofstoppingtimesτn,calledthelocalizingsequence,suchthatτn↑∞asn→∞andforeachnthestoppedprocessesX(t∧τn)hasthisproperty.Forexample,theuniformintegrabilitypropertyholdslocallyforanymartin-gale.ByTheorem7.13amartingaleconvergentinL1isuniformlyintegrable.HereM(t∧n)=M(n)fort>n,andthereforeτn=nisalocalizingsequence.Localmartingalesaredefinedbylocalizingthemartingaleproperty.Definition7.20AnadaptedprocessM(t)iscalledalocalmartingaleifthereexistsasequenceofstoppingtimesτn,suchthatτn↑∞andforeachnthestoppedprocessesM(t∧τn)isauniformlyintegrablemartingaleint.Aswehavejustseen,anymartingaleisalocalmartingale.Examplesoflocalmartingaleswhicharenotmartingalesaregivenbelow. 196CHAPTER7.MARTINGALESExample7.10:M(t)=1/|B(t)|,whereB(t)isthethree-dimensionalBrownianmotion,B(0)=x=0.WehaveseeninExample6.17thatifDristhecom-plementarysettotheballofradiusrcenteredattheorigin,Dr={z:|z|>r},thenf(z)=1/|z|isaharmonicfunctionfortheLaplacianonDr.Consequently1/|B(t∧τDr)|isamartingale,whereτDristhetimeofexitfromDr.TakenowτnbetheexittimefromD1/n,thatis,τn=inf{t>0:|B(t)|=1/n}.Thenforanyfixedn,1/|B(t∧τn)|isamartingale.τnincreaseto,say,τandbyconti-nuity,B(τ)=0.AsBrownianmotioninthreedimensionsnevervisitstheorigin(seeExample6.14),itfollowsbycontinuitythatτisinfinite.ThusM(t)isalocalmartingale.Toseethatitisnotatruemartingale,recallthatinthreedimensionsBrownianmotionistransientand|B(t)|→∞ast→∞.ThereforeEM(t)→0,whereasEM(0)=1/|x|.Sincetheexpectationofamartingaleisconstant,M(t)isnotamartingale.Example7.11:(Itˆointegrals.)t2B(s)LetM(t)=edB(s),t>1/4,whereBisBrownianmotioninonedimension0B2(t)withB(0)=0.Letτn=inf{t>0:e=n}.Thenfort≤τn,theintegrandisboundedbyn.BythemartingalepropertyofItˆointegrals,M(t∧τn)isamartingale2intforanyn.Bycontinuity,exp(B(τ))=∞,thusτn→τ=∞.ThereforeM(t)isalocalmartingale.Toseethatitisnotamartingalenoticethatfort>1/4,2B(t)Ee=∞,implyingthatM(t)isnotintegrable.Remark7.2:Notethatitisnotenoughforalocalmartingaletobeinte-grableinordertobeatruemartingale.Forexample,positivelocalmartingalesareintegrable,butingeneraltheyarenotmartingales,butonlysupermartin-gales(seeTheorem7.23)below.Evenuniformlyintegrablelocalmartingalesmaynotbemartingales.However,ifalocalmartingaleisdominatedbyanintegrablerandomvariablethenitisamartingale.Theorem7.21LetM(t),0≤t<∞,bealocalmartingalesuchthat|M(t)|≤Y,withEY<∞.ThenMisauniformlyintegrablemartingale.Proof:Letτnbealocalizingsequence.ThenforanynandsE(M2(0))unlessM(t)=M(0)a.s.ThusM2cannotbeamartingaleon[0,t],unlessM(t)=M(0).IfM(t)=M(0),thenforalls0.Conversely,if[M,M](t)=0,thenM(s)=0a.s.foralls≤t.Theresultalsoholdsforlocalmartingales.Proof:Weprovetheresultforsquareintegrablemartingales;forlocalmartingalesitcanbeshownbylocalization.Supposethat[M,M](t)=0forsomet>0.Then,since[M,M]isnon-decreasing,[M,M](s)=0foralls≤t.ByTheorem7.27M2(s),s≤t,isamartingale.Inparticular,E(M2(t))=0.ThisimpliesthatM(t)=0a.s.,whichisacontradiction.Therefore[M,M](t)>0.Conversely,if[M,M](t)=0,thesameargumentshowsthatM(t)=0a.s.,andbythemartingalepropertyM(s)=0foralls≤t.
ItalsofollowsfromtheproofthatMand[M,M]havesameintervalsofcon-stancy.Thistheoremimpliesremarkablythatacontinuousmartingalewhichisnotaconstanthasinfinitevariationonanyinterval.Theorem7.29LetMbeacontinuouslocalmartingale,andfixanyt.IfM(t)isnotidenticallyequaltoM(0),thenMhasinfinitevariationover[0,t].Proof:M(t)−M(0)isamartingale,nullatzero,withitsvalueattimetnotequalidenticallytozero.BytheabovetheoremMhasapositivequadraticvariationon[0,t],[M,M](t)>0.ByTheorem1.10acontinuousprocessoffi-nitevariationon[0,t]haszeroquadraticvariationoverthisinterval.ThereforeMmusthaveinfinitevariationover[0,t].
Corollary7.30Ifacontinuouslocalmartingalehasfinitevariationoveraninterval,thenitmustbeaconstantoverthatinterval.Remark7.3:Notethattherearemartingaleswithfinitevariation,butbythepreviousresulttheycannotbecontinuous.AnexampleofsuchamartingaleisthePoissonprocessmartingaleN(t)−t.7.7MartingaleInequalitiesM(t)denotesamartingaleoralocalmartingaleontheinterval[0,T]withpossiblyT=∞.Theorem7.31IfM(t)isamartingale(orapositivesubmartingale)thenforp≥1−ppP(sup|M(s)|≥a)≤asupE|M(s)|.(7.36)s≤ts≤t 7.7.MARTINGALEINEQUALITIES201Ifp>1,thenppEsup|M(s)|p≤E|M(t)|p.(7.37)s≤tp−1Thecaseofp=2iscalledDoob’sinequalityformartingales.EsupM(s)2≤4EM2(T).(7.38)s≤TAsaconsequence,ifforp>1,supE|M(t)|p<∞,thenM(t)ist≤Tuniformlyintegrable(ThisisaparticularcaseofTheorem7.7).Theorem7.32IfMislocallysquareintegrablemartingalewithM(0)=0,thenP(sup|M(t)|>a)≤a−2E([M,M](T)).(7.39)t≤TTheorem7.33(Davis’Inequality)Thereareconstantsc>0andC<∞suchthatforanylocalmartingaleM(t),nullatzero,)*cE[M,M](T)≤Esup|M(t)|≤CE[M,M](T).(7.40)t≤TTheorem7.34(Burkholder-GundyInequality)ThereareconstantscpandCpdependingonlyonp,suchthatforanylocalmartingaleM(t),nullatzero,)*cE[M,M](T)p/2≤E(sup|M(t)|)p≤CE[M,M](T)p/2,(7.41)ppt≤Tfor1t}.(7.47)If[M,M](t)isstrictlyincreasing,thenτtisitsinverse.Theorem7.37(Dambis,Dubins-Schwarz)LetM(t)beacontinuousmar-tingale,nullatzero,suchthat[M,M](t)isnon-decreasingto∞,andτtdefinedby(7.47).ThentheprocessB(t)=M(τt)isaBrownianmotionwithrespecttothefiltrationFτt.Moreover,[M,M](t)isastoppingtimewithrespecttothisfiltration,andthemartingaleMcanbeobtainedfromtheBrownianmotionBbythechangeoftimeM(t)=B([M,M](t)).TheresultalsoholdswhenMisacontinuouslocalmartingale.Weoutlinetheideaoftheproof,fordetailssee,forexample,RogersandWilliams(1990),p.64,KaratzasandShreve(1988)p.174,Protter(1992)p.81,RevuzandYor(1998)p.181.Proof:LetM(t)bealocalmartingale.τtdefinedby(7.47)arefinitestoppingtimes,since[M,M](t)→∞.ThusFτtarewelldefined,(seeChap-ter2forthedefinitionofFτ).Notethat{[M,M](s)≤t}={τt≥s}.Thisimpliesthat[M,M](s)arestoppingtimesforFτt.Since[M,M](s)iscontinuous[M,M](τt)=t.LetX(t)=M(τt).Thenitisacontinuouslo-calmartingale,sinceMand[M,M]havethesameintervalsofconstancy(seethecommentfollowingTheorem7.28).UsingTheorem7.27weobtainEX2(t)=E[X,X](t)=E[M,M](τ)=t.ThusXisaBrownianmotiontbyLevy’scharacterizationTheorem7.36.Thesecondpartisprovenasfol-lows.RecallthatMand[M,M]havethesameintervalsofconstancy.ThusX([M,M](t))=M(τ[M,M](t))=M(t).
tExample7.13:LetM(t)=f(s)dB(s),withfcontinuousandnon-random.0ThenMisaGaussianmartingale.Itsquadraticvariationisgivenby[M,M](t)=t2tf(s)ds.Forexample,withf(s)=s,M(t)=sdB(s)and[M,M](t)=00ts2ds=t3/3.Inthisexample[M,M](t)isnon-randomandincreasing.τtis0√33t1/3givenbyitsinverse,τt=(3t).LetX(t)=M(τt)=sdB(s).Then,clearly,0Xiscontinuous,asacompositionofcontinuousfunctions.Itisalsoamartingale3withquadraticvariationτt/3=t.Hence,bytheLevy’stheorem,itisaBrownian3motion,X(t)=Bˆ(t).Bytheabovetheorem,M(t)=Bˆ(t/3). 7.8.CONTINUOUSMARTINGALES.CHANGEOFTIME205tExample7.14:IfM(t)=H(s)dB(s)isanItˆointegral,thenitisalocalmar-0tingalewithquadraticvariation[M,M](t)=tH2(s)ds.If∞H2(s)ds=∞,then00t2M(t)=BˆH(s)ds,whereBˆ(t)isBrownianmotionandcanberecoveredfrom0M(t)withtheappropriatechangeoftime.Example7.15:(BrownianBridgeasTimeChangedBrownianmotion)t1TheSDEforBrownianBridge(5.34)containsasitsonlystochastictermdB(s).0T−sSinceforanyt0suchthattdsG(t)=0σ2(B(s))∞isfiniteforfinitet,andincreasestoinfinity,ds=∞almostsurely.0σ2(B(s))ThenG(t)isadapted,continuousandstrictlyincreasingtoG(∞)=∞.There-foreithasinverse(−1)τt=G(t).(7.55)Notethatforeachfixedt,τtisastoppingtime,asitisthefirsttimetheprocessG(s)hitst,andthatτtisincreasing.Theorem7.39TheprocessX(t)=B(τt)isaweaksolutiontotheSDEdX(t)=σ(X(t))dB(t).(7.56)Proof:X(t)=B(τ)=B(G(−1)(t)).Usingequation(7.52)withf=G(−1),tweobtain,dB(G(−1)(t))=(G(−1))(t)dBˆ(t). 7.8.CONTINUOUSMARTINGALES.CHANGEOFTIME207(−1)112(G)(t)===σ(B(τt)).(7.57)G(G(−1)(t))1/σ2(B(G(−1)(t)))Thusweobtain2dB(τt)=σ(B(τt))dBˆ(t),andtheresultisproved.
Anotherproof,byconsideringthemartingaleproblem,isgivennext.Notethat(7.57)givesdτ=σ2(B(τ))dt.ttProof:ThediffusionoperatorfortheSDE(7.56)isgivenbyLf(x)=1σ2(x)f(x).WeshowthatX(t)=B(τ)isasolutiontothemar-2ttingaleproblemforL.Indeed,weknow(seeExample5.17)thatforanytwicecontinuouslydifferentiablefunctionfvanishingoutsideacompactin-tterval,theprocessM(t)=f(B(t))−1f(B(s))dsisamartingale.Since02τtareincreasingstoppingtimesitcanbeshown(byusingOptionalStop-pingTheorem7.18)thattheprocessM(τt)isalsoamartingale,thatis,f(B(τ))−τt1f(B(s))dsisamartingale.Nowperformthechangeofvari-t02ables=τ,andobservefrom(7.57)thatdτ=σ2(B(τ))dt,toobtainthatutttheprocessf(B(τ))−t1σ2(B(τ))f(B(τ))duisamartingale.Butsincet02uuX(t)=B(τ),thisbeingthesameasf(X(t))−t1σ2(X(u))f(X(u))duisat02martingale,andX(t)solvesthemartingaleproblemforL.
AnapplicationofTheorem7.37givesaresultonuniquenessofthesolutionofSDE(7.56).ThisresultisweakerthanTheorem6.13ofEngelbert-Schmidt.Theorem7.40Letσ(x)beapositivefunctionboundedawayfromzero,σ(x)≥δ>0.Thenthestochasticdifferentialequation(7.56)hasauniqueweakso-lution.Proof:LetX(t)beaweaksolutionto(7.56).ThenX(t)isalocalmar-tingaleandthereisaBrownianmotionβ(t),suchthatX(t)=β([X,X](t)).Now,tt[X,X](t)=σ2(X(s))ds=σ2(β([X,X](s)))ds.00Thus[X,X](t)isasolutiontotheordinarydifferentialequation(ODE)da(t)=σ2(β(a(t))dt.SincethesolutiontothisODEisunique,thesolutionto(7.56)isunique.
Amoregeneralchangeoftimeisdoneforthestochasticdifferentialequa-tiondX(t)=µ(X(t))dt+σ(X(t))dB(t).(7.58)tLetg(x)beapositivefunctionforwhichG(t)=g(X(s))dsisfiniteforfinite0tandincreasestoinfinityalmostsurely.Defineτ=G(−1)(t).t 208CHAPTER7.MARTINGALESTheorem7.41LetX(t)beasolutionto(7.58)anddefineY(t)=X(τt).ThenY(t)isaweaksolutiontothestochasticdifferentialequationµ(Y(t))σ(Y(t))dY(t)=dt+dB(t),withY(0)=X(0).(7.59)g(Y(t))g(Y(t))Onecanusethechangeoftimeonaninterval[0,T],forastoppingtimeT.Example7.17:(Lamperti’sChangeofTime)LetX(t)satisfytheSDE(Feller’sbranchingdiffusion)dX(t)=µX(t)dt+σX(t)dB(t),X(0)=x>0,(7.60)twithpositiveconstantsµandσ.Lamperti’schangeoftimeisG(t)=X(s)ds.0Hereg(x)=x.ThenY(t)=X(τt)satisfiestheSDEµY(t)σY(t)dY(t)=dt+dB(t),Y(t)Y(t)=µdt+σdB(t)withY(0)=x,andY(t)=x+µt+σB(t).(7.61)Inotherwords,witharandomchangeoftime,theBranchingdiffusionisaBrownianmotionwithdrift.Atthe(random)pointwhereG(t)stopsincreasingitsinverseτt,definedastheright-inverseτt=inf{s:G(s)=t},alsoremainsthesame.ThishappensatthepointoftimewhenX(t)=0.Itcanbeseenthatoncetheprocessisatzero,itstaysat0forever.LetT=inf{t:X(t)=0}.Tisastoppingtime,andY(t)istheBrownianmotionstoppedatthattime.Theotherdirectionisalsotrue,aBranchingdiffusioncanbeobtainedfromaBrownianmotionwithdrift.LetY(t)satisfy(7.61),andandletT=inf{t:Y(t)=0}.Y(t)>0fort≤T.Definet∧T1G(t)=ds,Y(s)0andletτtbetheinverseofG,whichiswelldefinedon[0,T).ThenX(t)=Y(τt)satisfiestheSDE(7.60)stoppedwhenithitszero.Remark7.4:AnysolutiontoanSDEwithtimeindependentcoefficientscanbeobtainedfromBrownianmotionbyusingchangeofvariablesandrandomtimechange(GihmanandSkorohod(1972),p.113).TherearethreemainmethodsusedforsolvingSDEs:changeofstatespace,thatis,changeofvariable(Itˆo’sformula),changeoftimeandchangeofmeasure.WehaveseenexamplesofSDEssolvedbyusingchangeofvariables,andchangeoftime.Thechangeofmeasureapproachwillbecoveredlater.Notes.MaterialforthischapterisbasedonProtter(1992),RogersandWilliams(1990),GihmanandSkorohod(1972),LiptserandShiryayev(1977),(1989),RevuzandYor(1998). 7.9.EXERCISES2097.9ExercisesExercise7.1:LetM(t)beanFt-martingaleanddenoteitsnaturalfiltrationbyGt.ShowthatM(t)isaGt-martingale.Exercise7.2:Showthatanincreasingintegrableprocessisasubmartingale.Exercise7.3:ShowthatifX(t)isasubmartingaleandgisanon-decreasingconvexfunctionsuchthatE|g(X(t))|<∞,theng(X(t))isasubmartingale.Exercise7.4:ShowthatM(t)isasquareintegrablemartingaleifandonlyifM(t)=E(Y|F),whereYissquareintegrable,E(Y2)<∞.tExercise7.5:(ExpectedexittimeofBrownianmotionfrom(a,b).)LetB(t)beaBrownianmotionstartedatx∈(a,b),andτ=inf{t:B(t)=aorb}.BystoppingthemartingaleM(t)=B(t)2−t,showthatE(τ)=x(x−a)(b−x).Exercise7.6:FindtheprobabilityofB(t)−t/2reachingabeforeitreachesbwhenstartedatx,a0suchthatEe−R(c−X1)=1.Showthatforalln,P(T≤n)≤e−Rx,whereU=xtheinitialfunds,andtheruinprobabilityx0P(T<∞)≤e−Rx.Hint:showthatM=e−RUnisamartingale,andusexntheOptionalStoppingTheorem.Exercise7.10:(RuinProbabilityinInsurancecontinued)FindtheboundontheruinprobabilitywhentheaggregateclaimshaveN(µ,σ2)distribution.Givetheinitialamountxrequiredtokeeptheruinprobabilitybelowlevelα. 210CHAPTER7.MARTINGALESExercise7.11:LetB(t)beaBrownianmotionstartingatzeroandTbethefirstexittimefrom(−1,1),thatis,thefirsttimewhen√|B|takesvalue1.UseDavis’inequalitytoshowthatE(T)<∞.tExercise7.12:LetB(t)beaBrownianmotion,X(t)=sign(B(s))dB(s).0ShowthatXisalsoaBrownianmotion.tExercise7.13:LetM(t)=esdB(s).Findg(t)suchthatM(g(t))isa0Brownianmotion.Exercise7.14:LetB(t)beaBrownianmotion.GiveanSDEfore−αtB(e2αt).Exercise7.15:ProvethechangeoftimeresultinSDEs,Theorem7.41.Exercise7.16:LetX(t)satisfySDEdX(t)=µ(t)dt+σ(t)dB(t)on[0,T].ShowthatX(t)isalocalmartingaleifandonlyifµ(t)=0a.e.Exercise7.17:f(x,t)isdifferentiableintandtwiceinx.ItisknownthatX(t)=f(B(t),t)isoffinitevariation.Showthatfisafunctionoftalone.ttExercise7.18:LetY(t)=B(s)dB(s)andW(t)=sign(B(s))dB(s).00ShowthatdY(t)=t+2Y(t)dW(t).ShowuniquenessoftheweaksolutionoftheaboveSDE. Chapter8CalculusForSemimartingalesInthischapterrulesofcalculusaregivenforthemostgeneralprocessesforwhichstochasticcalculusisdeveloped,calledsemimartingales.Asemimartin-galeisprocessconsistingofasumofalocalmartingaleandafinitevariationprocess.Integrationwithrespecttosemimartingalesinvolvesintegrationwithrespecttolocalmartingales,andtheseintegralsgeneralizetheItˆointegralwhereintegrationisdonewithrespecttoaBrownianmotion.Importantcon-cepts,suchascompensatorsandthesharpbracketprocessesareintroduced,andItˆo’sformulainitsgeneralformisgiven.8.1SemimartingalesInstochasticcalculusonlyregularprocessesareconsidered.Theseareeithercontinuousprocesses,orright-continuouswithleftlimits,orleft-continuouswithrightlimits.Theregularityoftheprocessimpliesthatitcanhaveatmostcountablymanydiscontinuities,andallofthemarejumps(Chapter1).Thedefinitionofasemimartingalepresumesagivenfiltrationandprocesseswhichweconsiderareadaptedtoit.Followingtheclassicalapproach,seeforexample,Metivier(1982),LiptserandShiryayev(1989)p.85,asemimartingale,isalocalmartingaleplusaprocessoffinitevariation.Moreprecisely,Definition8.1Aregularright-continuouswithleftlimits(c`adl`ag)adaptedprocessisasemimartingaleifitcanberepresentedasasumoftwoprocesses:alocalmartingaleM(t)andaprocessoffinitevariationA(t),withM(0)=A(0)=0,andS(t)=S(0)+M(t)+A(t).(8.1)211 212CHAPTER8.CALCULUSFORSEMIMARTINGALESExample8.1:(Semimartingales)21.S(t)=B(t),whereB(t)isaBrownianmotionisasemimartingale.S(t)=2M(t)+t,whereM(t)=B(t)−tisamartingaleandA(t)=tisafinitevariationprocess.2.S(t)=N(t),whereN(t)isaPoissonprocesswithrateλ,isasemimartingale,asitisafinitevariationprocess.3.Onewaytoobtainsemimartingalesfromknownsemimartingalesisbyap-2plyingatwicecontinuouslydifferentiable(C)transformation.IfS(t)isa2semimartingaleandfisaCfunction,thenf(S(t))isalsoasemimartingale.Thedecompositionoff(S(t))intomartingalepartandfinitevariationpartisgivenbyItˆo’sformula,givenlater.Inthiswaywecanassertthat,forexample,σB(t)+µtthegeometricBrownianmotioneisasemimartingale.4.Aright-continuouswithleftlimits(c`adl`ag)deterministicfunctionf(t)isasemimartingaleifandonlyifitisoffinitevariation.Thusf(t)=tsin(1/t),t∈(0,1],f(0)=0iscontinuous,butnotasemimartingale(seeExample1.7).5.Adiffusion,thatis,asolutiontoastochasticdifferentialequationwithrespecttoBrownianmotion,isasemimartingale.Indeed,theItˆointegralwithrespecttodB(t)isalocalmartingaleandtheintegralwithrespecttodtisaprocessoffinitevariation.6.Althoughtheclassofsemimartingalesisratherlarge,thereareprocesseswhichαarenotsemimartingales.Examplesare:|B(t)|,0<α<1,whereB(t)istheone-dimensionalBrownianmotion;t(t−s)−αdB(s),0<α<1/2.Itrequires0analysistoshowthattheaboveprocessesarenotsemimartingales.ForasemimartingaleX,theprocessofjumps∆Xisdefinedby∆X(t)=X(t)−X(t−),(8.2)andrepresentsthejumpatpointt.IfXiscontinuous,thenofcourse,∆X=0.8.2PredictableProcessesInthissectionwedescribetheclassofpredictableprocesses.Thisclassofprocesseshasacentralroleinthetheory.Inparticular,onlypredictableprocessescanbeintegratedwithrespecttoasemimartingale.RecallthatindiscretetimeaprocessHispredictableifHnisFn−1measurable,thatis,Hisknownwithcertaintyattimenonthebasisofinformationuptotimen−1.Predictabilityincontinuoustimeishardertodefine.Werecallsomegeneraldefinitionsofprocessesstartingwiththeclassofadaptedprocesses.Definition8.2AprocessXiscalledadaptedtofiltrationIF=(Ft),ifforallt,X(t)isFt-measurable. 8.2.PREDICTABLEPROCESSES213tInconstructionofthestochasticintegralH(u)dS(u),processesHandSare0takentobeadaptedtoIF.ForageneralsemimartingaleS,therequirementthatHisadaptedistooweak,itfailstoassuremeasurabilityofsomebasicconstructions.Hmustbepredictable.Theexactdefinitionofpredictable+processesinvolvesσ-fieldsgeneratedonIR×ΩandisgivenlaterinSection8.13.Notethatleft-continuousprocessesarepredictable,inthesensethatH(t)=lims↑tH(s)=H(s−).Sothatifthevaluesoftheprocessbeforetareknown,thenthevalueattisdeterminedbythelimit.Forourpurposesitisenoughtodescribeasubclassofpredictableprocesseswhichcanbedefinedconstructively.Definition8.3Hispredictableifitisoneofthefollowing:a)aleft-continuousadaptedprocess,inparticular,acontinuousadaptedprocess.b)alimit(almostsure,inprobability)ofleft-continuousadaptedprocesses.c)aregularright-continuousprocesssuchthat,foranystoppingtimeτ,HτisFτ−-measurable,theσ-fieldgeneratedbythesetsA∩{Tt}∈Ft.ConsidertheprocessX(t)=I[0,T](t).Itisadapted,becauseitsvaluesaredeterminedbytheset{T≤t}(X(t)=1ifandonlyifω∈{T≤t}),cand{T≤t}={T>t}∈Ft.X(t)isalsoleft-continuous.Thusitisapredictableprocessbya).WealsoseethatTisastoppingtimeifandonlyiftheprocessX(t)=I[0,T](t)isadapted.Example8.5:ItwillbeseenlaterthatwhenfiltrationisgeneratedbyBrownianmotion,thenanyright-continuousadaptedprocessispredictable.ThisiswhyinthedefinitionoftheItˆointegralright-continuousfunctionsareallowedasintegrands. 214CHAPTER8.CALCULUSFORSEMIMARTINGALES8.3Doob-MeyerDecompositionRecallthataprocessisasubmartingaleifforalls0andsign(x)=−1forx≤0.ItispossibletoextendItˆo’sformulaforthiscaseandprove(seeRogersandWilliams(1990),p.95-102,Protter(1992)p.165-167)Theorem8.9(Tanaka’sFormula)LetX(t)beacontinuoussemimartin-gale.Thenforanya∈IRthereexistsacontinuousnon-decreasingadaptedprocessLa(t),calledthelocaltimeataofX,suchthatt|X(t)−a|=|X(0)−a|+sign(X(s)−a)dX(s)+La(t).(8.29)0Asafunctionina,La(t)isright-continuouswithleftlimits.ForanyfixedaasafunctionintLa(t)increasesonlywhenX(t)=a,thatis,La(t)=tI(X(s)=a)dLa(s).Moreover,ifX(t)isacontinuouslocalmartingale,0thenLa(t)isjointlycontinuousinaandt.Remark8.2:HeuristicallyTanaka’sformulacanbejustifiedbyaformalapplicationofItˆo’sformulatothefunctionsign(x).Thederivativeofsign(x)iszeroeverywherebutatzero,whereitisnotdefined.However,itispossibletodefinethederivativeasageneralizedfunctionoraSchwartzdistribution,inwhichcaseitisequalto2δ.Thusthesecondderivativeof|x−a|isδ(x−a)inthegeneralizedfunctionsense.ThelocaltimeataofXisdefinedastLa(t)=δ(X(s)−a)ds.FormaluseofItˆo’sformulagives8.29.0Theorem8.10(OccupationTimesFormula)LetX(t)beacontinuoussemimartingalewithlocaltimeLa(t).Thenforanyboundedmeasurablefunc-tiong(x)t∞g(X(s))d[X,X](s)=g(a)La(t)da.(8.30)0−∞Inparticular∞[X,X](t)=La(t)da.(8.31)−∞Example8.10:LetX(t)=B(t)beBrownianmotion.Thenitslocaltimeatzero0process,L(t)satisfies(Tanaka’sformula)t0L(t)=|B(t)|−sign(B(s))dB(s).(8.32)0 8.7.LOCALTIMES223Theoccupationtimesformula(8.30)becomest∞ag(B(s))ds=g(a)L(t)da.(8.33)0−∞ThetimeBrownianmotionspendsinasetA⊂IRuptotimetisgivenby(withg(x)=IA(x))t∞aaIA(B(s))ds=IA(a)L(t)da=L(t)da.(8.34)0−∞ARemark8.3:TakingA=(a,a+da)andg(x)=I(a,a+da)(x)itsindicatorin(8.33),La(t)daisthetimeBrownianmotionspendsin(a,a+da)uptotimet,whichexplainsthename“localtime”.ThetimeBrownianmotionspendsinasetAisLa(t)da,thereforethename“occupationtimesdensity”formulaA(8.34).Foracontinuoussemimartingaletheformula(8.30)isthe“occupationtimesdensity”formularelativetotherandom“clock”d[X,X](s).Example8.11:X(t)=|B(t)|isasemimartingale,since|x|isaconvexfunction.ItsdecompositionintothemartingaleandfinitevariationpartsisgivenbyTanaka’sformula(8.32).|B(t)|isnotasemimartingale,seeProtter(1992),p.169-170.+Example8.12:Thefunction(x−a)isimportantinfinancialapplication,asit+givesthepayoffofafinancialstockoption.TheMeyer-Tanaka’sformulafor(x−a)t++1a(X(t)−a)=(X(0)−a)+I(X(s)>a)dX(s)+Lt.(8.35)20Theorem8.11LetLa(t)bethelocaltimeofBrownianmotionata,andf(a)tthedensityofN(0,t)ata.ThentaadE(L(t))E(L(t))=fs(a)dahence=ft(a).(8.36)0dtProof:Takingexpectationinbothsidesofequation(8.33)andchangingtheorderofintegration,weobtainforanypositiveandboundedg∞∞g(a)f(a)dsda=g(a)E(La(t))da.s−∞−∞Theresultfollows,sincegisarbitrary.
Asimilarresultcanbeestablishedforcontinuoussemimartingalesbyusingequation(8.30)(e.g.Klebaner(2002)).Remark8.4:Localtimescanalsobedefinedfordiscontinuoussemimartin-gales.Foranyfixeda,La(t)isacontinuousnon-decreasingfunctionint,anditincreasesonlyatpointsofcontinuityofXwhereitisequaltoa,thatis,X(t−)=X(t)=a.Theformula(8.30)holdswithquadraticvariation[X,X]replacedbyitscontinuouspart[X,X]c,seeforexample,Protter(1992),p.168. 224CHAPTER8.CALCULUSFORSEMIMARTINGALES8.8StochasticExponentialThestochasticexponential(alsoknownasthesemimartingale,orDol´eans-Dadeexponential)isastochasticanalogueoftheexponentialfunction.Recallthatiff(t)isasmoothfunctiontheng(t)=ef(t)isthesolutiontothedif-ferentialequationdg(t)=g(t)df(t).Thestochasticexponentialisdefinedasasolutiontoasimilarstochasticequation.ThestochasticexponentialofItˆoprocesseswasintroducedinSection5.2.ForasemimartingaleX,itsstochasticexponentialE(X)(t)=U(t)isdefinedastheuniquesolutiontotheequationtU(t)=1+U(s−)dX(s)ordU(t)=U(t−)dX(t);withU(0)=1.(8.37)0AsanapplicationofItˆo’sformulaandtherulesofstochasticcalculusweproveTheorem8.12LetXbeacontinuoussemimartingale.ThenitsstochasticexponentialisgivenbyX(t)−X(0)−1[X,X](t)U(t)=E(X)(t)=e2.(8.38)Proof:WriteU(t)=eV(t),withV(t)=X(t)−X(0)−1[X,X](t).Then2V(t)V(t)1V(t)dU(t)=d(e)=edV(t)+ed[V,V](t).2Usingthefactthat[X,X](t)isacontinuousprocessoffinitevariation,weobtain[X,[X,X]](t)=0,and[V,V](t)=[X,X](t).Usingthis,weobtainV(t)1V(t)1V(t)V(t)dU(t)=edX(t)−ed[X,X](t)+ed[X,X](t)=edX(t),22ordU(t)=U(t)dX(t).ThusU(t)definedby(8.38)satisfies(8.37).Toshowuniqueness,letV(t)beanothersolutionto(8.37),andconsiderV(t)/U(t).ByintegrationbypartsV(t)111d()=V(t)d()+dV(t)+d[V,](t).U(t)U(t)U(t)UByItˆo’sformula,usingthatU(t)iscontinuousandsatisfies(8.37)111d()=−dX(t)+d[X,X](t),U(t)U(t)U(t)whichleadstoV(t)V(t)V(t)V(t)V(t)d()=−dX(t)+dX(t)+d[X,X](t)−d[X,X](t)=0.U(t)U(t)U(t)U(t)U(t)ThusV(t)/U(t)=const.=V(0)/U(0)=1.
Propertiesofthestochasticexponentialaregivenbythefollowingresult. 8.8.STOCHASTICEXPONENTIAL225Theorem8.13LetXandYbesemimartingalesonthesamespace.Then1.E(X)(t)E(Y)(t)=E(X+Y+[X,Y])(t)2.IfXiscontinuous,X(0)=0,then(E(X)(t))−1=E(−X+[X,X])(t).Theproofusestheintegrationbypartsformulaandisleftasanexercise.Example8.13:(StockprocessanditsReturnprocess.)Anapplicationinfinanceisprovidedbytherelationbetweenthestockprocessanditsreturnprocess.ThereturnisdefinedbydR(t)=dS(t)/S(t−).Hencethestockpriceisthestochasticexponentialofthereturn,dS(t)=S(t−)dR(t),andS(t)=S(0)E(R)(t).StochasticExponentialofMartingalesStochasticexponentialU=E(M)ofamartingale,oralocalmartingale,M(t)isastochasticintegralwithrespecttoM(t).Sincestochasticintegralswithrespecttomartingalesorlocalmartingalesarelocalmartingales,E(M)isalocalmartingale.InapplicationsitisimportanttohaveconditionsforE(M)tobeatruemartingale.Theorem8.14(Martingaleexponential)LetM(t),0≤t≤T<∞beacontinuouslocalmartingalenullatzero.ThenitsstochasticexponentialE(M)M(t)−1[M,M](t)isgivenbye2anditisacontinuouspositivelocalmartingale.Consequently,itisasupermartingale,itisintegrableandhasafinitenon-increasingexpectation.Itisamartingaleifanyofthefollowingconditionshold.1M(T)−[M,M](T)1.Ee2=1.2.Forallt≥0,Ete2M(s)−[M,M](s)d[M,M](s)<∞.03.Forallt≥0,Ete2M(s)d[M,M](s)<∞.0Moreover,iftheexpectationsaboveareboundedbyK<∞,thenE(M)isasquareintegrablemartingale.M(t)−1[M,M](t)Proof:ByTheorem8.12,E(M)(t)=e2,thereforeitispos-itive.Beingastochasticintegralwithrespecttoamartingale,itisalocalmartingale.ThusE(M)isasupermartingale,asapositivelocalmartingale,seeTheorem7.23.Asupermartingalehasanon-increasingexpectation,andisamartingaleifandonlyifitsexpectationatTisthesameasat0(seeTheorem7.3).Thisgivesthefirstcondition.ThesecondconditionfollowsfromTheo-rem7.35,whichstatesthatifalocalmartingalehasfiniteexpectationofits 226CHAPTER8.CALCULUSFORSEMIMARTINGALESquadraticvariation,thenitisamartingale.SothatifE[E(M),E(M)](t)<∞,thenE(M)isamartingale.Bythequadraticvariationofanintegralt[E(M),E(M)](t)=e2M(s)−[M,M](s)d[M,M](s).(8.39)0Thethirdconditionfollowsfromthesecond,since[M,M]ispositiveandincreasing.ThelaststatementfollowsbyTheorem7.35,sincetheboundimpliessup0≤tE[E(M),E(M)](t)<∞.
Theorem8.15(Kazamaki’scondition)LetMbeacontinuouslocalmar-1M(t)tingalewithM(0)=0.Ife2isasubmartingale,thenE(M)isamartin-gale.TheresultisproveninRevuzandYor(1999),p.331.Theorem8.16LetMbeacontinuousmartingalewithM(0)=0.If1M(T)Ee2<∞,(8.40)thenE(M)isamartingaleon[0,T].Proof:ByJensen’sinequality(seeExercise(7.3))ifgisaconvexfunction,andE|g(M(t)|<∞fort≤T,thenEg(M(t))≤Eg(M(T))andg(M(t))isasubmartingale.Sinceex/2isconvex,theresultfollowsbyKazamaki’scondition.
Theorem8.17(Novikov’scondition)LetMbeacontinuouslocalmar-tingalewithM(0)=0.Supposethatforeacht≤T1[M,M](t)Ee2<∞.(8.41)ThenE(M)isamartingalewithmeanone.Inparticular,ifforeachtthereisaconstantKtsuchthat[M,M](t)0.5,whichgiveafinitenon-zerocontributiontotheproduct.Takingtheproductwithoversatwhich|∆X(s)|≤1/2,andtakinglogarithm,itisenoughtoshowthat|ln(1+∆X(s))−∆X(s)|converges.Butthiss≤tfollowsfromtheinequality|ln(1+∆X(s))−∆X(s)|≤(∆X(s))2by(8.51).ToseethatU(t)definedby(8.62)satisfies(8.61)useItˆo’sformulaappliedtothefunctionf(Y(t),V(t))withf(x,x)=ex1x.Fortheuniquenessofthe122solutionof(8.61)andotherdetailsseeLiptserandShiryayev(1989),p.123.
Example8.23:Thestochasticexponential(8.62)ofaPoissonprocessiseasilyN(t)seentobeE(N)(t)=2.IfU=E(X)isthestochasticexponentialofX,thenX=L(U)isthestochas-ticlogarithmofU,satisfyingequation(8.61)dU(t)dX(t)=,orL(E(X))=X.U(t−)ForItˆoprocessesanexpressionforX(t)isgiveninTheorem5.3,forgeneralcaseseeExercise8.17. 8.12.MARTINGALE(PREDICTABLE)REPRESENTATIONS2378.12Martingale(Predictable)RepresentationsInthissectionwegiveresultsonrepresentationofmartingalesbystochasticintegralsofpredictableprocesses,alsocalledpredictablerepresentations.LetM(t)beamartingale,0≤t≤T,adaptedtothefiltrationIF=(Ft),andH(t)TbeapredictableprocesssatisfyingH2(s)dM,M(s)<∞withprobability0tone.ThenH(s)dM(s)isalocalmartingale.Thepredictablerepresentation0propertymeansthattheconverseisalsotrue.LetIFM=(FM)denotethetnaturalfiltrationofM.Definition8.34AlocalmartingaleMhasthepredictablerepresentationprop-MertyifforanyIF-localmartingaleXthereisapredictableprocessHsuchthattX(t)=X(0)+H(s)dM(s).(8.64)0Thisdefinitionisdifferenttotheclassicaloneformartingaleswithjumps,seeRemark8.7below,butisthesameforcontinuousmartingales.Brownianmotionhasthepredictablerepresentationproperty(see,forex-ampleRevuzandYor(1999)p.209,LiptserandShiryayev(2001)Ip.170).Theorem8.35(BrownianMartingaleRepresentation)LetX(t),0≤t≤T,bealocalmartingaleadaptedtotheBrownianfil-BtrationIF=(Ft).ThenthereexistsapredictableprocessH(t)suchthatTH2(s)ds<∞withprobabilityone,andequation(8.65)holds0tX(t)=X(0)+H(s)dB(s).(8.65)0Moreover,ifYisanintegrableFT-measurablerandomvariable,E|Y|<∞,thenTY=EY+H(t)dB(t).(8.66)0Ifinaddition,YandBhavejointlyaGaussiandistribution,thentheprocessH(t)in(8.66)isdeterministic.Proof:Wedon’tprovetherepresentationofamartingale,butonlytherepresentationforarandomvariablebasedonit.TakeX(t)=E(Y|Ft).ThenX(t),0≤t≤T,isamartingale(seeTheorem7.9).HencebythemartingalerepresentationthereexistsH,suchthattX(t)=X(0)+H(s)dB(s).Takingt=Tgivestheresult.0
238CHAPTER8.CALCULUSFORSEMIMARTINGALESRemark8.6:AfunctionalofthepathoftheBrownianmotionB[0,T]isarandomvariableY,FT-measurable.Theorem8.35statesthatundertheaboveassumptions,anyfunctionalofBrownianmotionhastheform(8.66).SinceItˆointegralsarecontinuous,andanylocalmartingaleofaBrownianfiltrationisanItˆointegral,itfollowsthatalllocalmartingalesofaBrownianfiltrationarecontinuous.Infactwehavethefollowingresult(thesecondstatementisnotstraightforward)Corollary8.361.AlllocalmartingalesoftheBrownianfiltrationarecontinuous.2.Allright-continuousadaptedprocessesarepredictable.Corollary8.37LetX(t),0≤t≤T,beasquareintegrablemartingaleadaptedtotheBrownianfiltrationIF.ThenthereexistsapredictableprocessH(t)suchthatTEH2(s)ds<∞andrepresentation(8.65)holds.Moreover,0tdX,B(t)X,B(t)=H(s)ds,andH(t)=.(8.67)0dtTheequation(8.67)followsfrom(8.65)bytheruleofthesharpbracketforintegrals.Example8.24:(Representationofmartingales)2t1.X(t)=B(t)−t.ThenX(t)=2B(s)dB(s).HereH(t)=2B(t),whichcan0alsobefoundbyusing(8.67).∂f2.LetX(t)=f(B(t),t)beamartingale.ByItˆo’sformuladX(t)=(B(t),t)dB(t).∂x∂fThusH(t)=(B(t),t).Thisalsoshowsthat∂xf(B,t),B(t)∂f=(B(t),t).dt∂xExample8.25:(Representationofrandomvariables)TT1.IfY=B(s)ds,thenY=(T−s)dB(s).002222.Y=B(1).ThenM(t)=E(B(1)|Ft)=B(t)+(1−t).UsingItˆo’sformulafor21M(t)weobtainB(1)=1+2B(t)dB(t).0SimilarresultsholdforthePoissonprocessfiltration.Theorem8.38(PoissonMartingaleRepresentation)LetM(t),0≤t≤T,bealocalmartingaleadaptedtothePoissonfiltration.ThenthereexistsapredictableprocessH(t)suchthattM(t)=M(0)+H(s)dN¯(s),(8.68)0whereN¯(t)=N(t)−tisthecompensatedPoissonprocess. 8.12.MARTINGALE(PREDICTABLE)REPRESENTATIONS239Whenafiltrationislargerthanthenaturalfiltrationofamartingale,thenthereisthefollowingresult(RevuzandYor(1999)p.209,LiptserandShiryaev(2001)p.170).Theorem8.39IfM(t),0≤t≤T,isanycontinuouslocalmartingale,andMXacontinuousIF-localmartingale.ThenXhasarepresentationtX(t)=X(0)+H(s)dM(s)+Z(t),(8.69)0whereHispredictableandM,Z=0;(consequentlyX−Z,Z=0).Example8.26:LetIFbegeneratedbytwoindependentBrownianmotionsBtandW,andletM(t)=W(s)dB(s).Itisamartingale,asastochasticintegral0t2satisfyingEW(s)ds<∞.WeshowthatMdoesnothavethepredictable0representationproperty.M,M(t)=tW2(s)ds.HenceW2(t)=dS,S(t),whichshowsthatW2(t)isFM-0dtt2Mmeasurable.HencethemartingaleX(t)=W(t)−tisadaptedtoFt,butisnotanintegralofapredictableprocesswithrespecttoM.ByItˆo’sformulatt2X(t)=2W(s)dW(s).HenceX,M(t)=W(s)dW,B(s)=0.Suppose00tthereisH,suchthatX(t)=H(u)dM(u).Thenby(8.67)0dX,M(t)H(t)==0,implyingthatX(t)=0,whichisacontradiction.dtThisexamplehasanapplicationinFinance,itshowsnon-completenessofastochasticvolatilitymodel.Example8.27:LetIFbegeneratedbyaBrownianmotionsBandaPoissonprocessN,andletM(t)=B(t)+N(t)−t=B(t)+N¯(t),whereN¯(t)=N(t)−t.Misamartingale,asasumoftwomartingales.WeshowthatMdoesnothavethepredictablerepresentationproperty.[M,M](t)=[B,B](t)+[N,N](t)=t+N(t).ThisshowsthatN(t)=[M,M](t)−tMMisFt-measurable.HenceB(t)=M(t)−N(t)+tisFt-measurable.ThusthetMmartingaleX(t)=N(s−)dB(s)isFt-measurable,butitdoesnothaveapre-0tttdictablerepresentation.Ifitdid,N(s−)dB(s)=H(s)dB(s)+H(s)dN¯(s),000ttand(N(s−)−H(s))dB(s)=H(s)dN¯(s).Sincetheintegralontherhsisof00tfinitevariation,H(s)=N(s−),foralmostalls.ThusH(s)dN¯(s)=0.Thisis0ttthesameasN(s−)dN(s)=N(s−)ds.Butthisisimpossible.Toseethecon-00T2tradiction,lett=T2thetimeofthesecondjumpofN.ThenN(s−)dN(s)=10T2andN(s−)ds=T2−T1.0Thisexampleshowsnon-completenessofmodelsofstockpriceswithjumps.Itcanbegeneralizedtoamodelthatsupportsamartingalewithajumpcomponent.Remark8.7:TheDefinition8.34ofthepredictablerepresentationpropertygivenhereagreeswiththestandarddefinitiongivenforcontinuousmartingales 240CHAPTER8.CALCULUSFORSEMIMARTINGALESbutisdifferenttothedefinitionforpredictablerepresentationwithrespecttosemimartingales,giveninLiptserandShiryaev(1989)p.250,JacodandShiryaev(1987)p.172,Protter(1992)p.150.ThegeneraldefinitionallowsfordifferentpredictablefunctionshandHtobeusedintheintegralswithrespecttothecontinuousmartingalepartMcandthediscretemartingalepartMdofM,ttX(t)=X(0)+h(s)dMc(s)+H(s)dMd(s).00Inthisdefinition,themartingaleMinExample8.27hasthepredictablerep-resentationproperty.Thedefinitiongivenhereismoresuitableforfinancialapplications.Ac-cordingtothefinancialmathematicstheoryanoptioncanbepricedifitcanbereplicated,whichmeansthatitisanintegralofapredictableprocessHwithrespecttothediscountedstockpriceprocessM,whichisamartingale.TheprocessHrepresentsthenumberofsharesbought/soldsoitdoesnotmakesensetohaveHconsistoftwodifferentcomponents.8.13ElementsoftheGeneralTheoryThebasicsetupconsistsoftheprobabilityspace(Ω,F,P),whereFisaσ-field+onΩandPisaprobabilityonF.AstochasticprocessisamapfromIR×Ω+toIR,namely(t,ω)→X(t,ω).IRhastheBorelσ-fieldofmeasurablesets,andFistheσ-fieldofmeasurablesetsonΩ.Onlymeasurableprocessesareconsidered,thatisforanyA∈B+{(t,ω):X(t,ω)∈A}∈B(IR)×F.Theorem8.40(Fubini)LetX(t)beameasurablestochasticprocess.Then1.P-a.s.thefunctionsX(t,ω)(trajectories)are(Borel)measurable.2.IfEX(t)existsforallt,thenitismeasurableasafunctionoft.b3.IfE|X(t)|dt<∞P-a.s.thenalmostalltrajectoriesX(t)areinte-abbgrableandEX(t)dt=EX(t)dt.aaLetIFbeafiltrationofincreasingσ-fieldsonΩ.Importantclassesofprocessesareintroducedviameasurabilitywithrespecttovariousσ-fieldsofsubsetsof+IR×Ω:adapted,progressivelymeasurableprocesses,optionalprocessesandpredictableprocesses,givenintheorderofinclusion. 8.13.ELEMENTSOFTHEGENERALTHEORY241Xisadaptedif,forallt,X(t)isFtmeasurable.Xisprogressivelymeasur-ableif,foranyt,{(s≤t,ω):X(s,ω)∈A}∈B([0,t])×Ft.Anyprogressivelymeasurableprocessisclearlyadapted.Itcanbeshownthatanyrightorleftcontinuousprocessisprogressivelymeasurable.Definition8.411.Theσ-fieldgeneratedbytheadaptedleft-continuousprocessesiscalledthepredictableσ-fieldP.2.Theσ-fieldgeneratedbytheadaptedright-continuousprocessesiscalledtheoptionalσ-fieldO.3.Aprocessiscalledpredictableifitismeasurablewithrespecttothepre-dictableσ-fieldP;itiscalledoptionalifitismeasurablewithrespecttotheoptionalσ-fieldO.Remarks1.Thepredictableσ-fieldPisalsogeneratedbytheadaptedcontinuousprocesses.2.DefineFt−=σ∪st},whereA∈Ft,t>0.Therearethreetypesofstoppingtimesthatareusedinstochasticcalculus:1.predictablestoppingtimes,2.accessiblestoppingtimes,3.totallyinaccessiblestoppingtimes.τisapredictablestoppingtimeifthereexistsasequenceofstoppingtimesτn,τn<τ,andlimnτn=τ.Inthiscaseitissaidthatthesequenceτnannouncesτ.Example8.29:Ifτisastoppingtimethenforanyconstanta>0,τ+aisapredictablestoppingtime.Indeed,itcanbeapproachedbyτn=τ+a−1/n.Example8.30:LetB(t)beBrownianmotionstartedatzero,IFitsnaturalfiltra-tionandτthefirsthittingtimeof1,thatis,τ=inf{t:B(t)=1}.τisapredictablestoppingtime,sinceτn=inf{t:B(t)=1−1/n}convergetoτ. 8.13.ELEMENTSOFTHEGENERALTHEORY243τisanaccessiblestoppingtimeifitispossibletoannounceτ,butwithdif-ferentsequencesondifferentpartsofΩ,thatis,[[τ]]⊂∪n[[τn]],whereτnarepredictablestoppingtimes.Allothertypesofstoppingtimesarecalledtotallyinaccessible.Example8.31:LetN(t)bePoissonprocess,IFitsnaturalfiltrationandτthetimeofthefirstjump,τ=inf{t:N(t)=1}.τisatotallyinaccessiblestoppingtime.Anypredictablestoppingtimeτn<τisaconstant,sinceFt∩{t<τ}istrivial.Butτhasacontinuousdistribution(exponential),thusitcannotbeapproachedbyconstants.Theoptionalσ-fieldisgeneratedbythestochasticintervals[[0,τ[[,whereτisastoppingtime.Thepredictableσ-fieldisgeneratedbythestochasticintervals[[0,τ]].AsetAiscalledpredictableifitsindicatorisapredictableprocess,IA∈P.Thefollowingresultsallowustodecideonpredictability.Theorem8.42LetX(t)beapredictableprocessandτbeastoppingtime.Then1.X(τ)I(τ<∞)isFτ−measurable,2.thestoppedprocessX(t∧τ)ispredictable.Foraproofsee,forexample,LiptserandShiryayev(1989)p.13.Theorem8.43AnadaptedregularprocessispredictableifandonlyifforanypredictablestoppingtimeτtherandomvariableX(τ)I(τ<∞)isFτ−measurableandforeachtotallyinaccessiblestoppingtimeτoneofthefollowingtwoconditionshold1.X(τ)=X(τ−)onτ<∞2.theset{∆X=0}∩[[τ]]isP-evanescent.Foraproofsee,forexample,LiptserandShiryayev(1989)p.16.Theorem8.44AstoppingtimeτispredictableifandonlyifforanyboundedmartingaleM,EM(τ)I(τ<∞)=EM(τ−)I(τ<∞).Theorem8.45ThecompensatorA(t)iscontinuousifandonlyifthejumptimesoftheprocessX(t)aretotallyinaccessible.See,forexample,LiptserandShiryayev(2001)fortheproof.Example8.32:ThecompensatorofthePoissonprocessist,whichiscontinuous.BytheaboveresultthejumptimesofthePoissonprocessaretotallyinaccessible.ThiswasshowninExample8.31. 244CHAPTER8.CALCULUSFORSEMIMARTINGALESItcanbeshownthatforBrownianmotionanddiffusionsanystoppingtimeispredictable.Thisimpliesthattheclassofoptionalprocessesisthesameastheclassofpredictableprocesses.Theorem8.46ForBrownianmotionfiltrationanymartingale(localmartin-gale)iscontinuousandanypositivestoppingtimeispredictable.Anyoptionalprocessisalsopredictable,O=P.Similarresultholdsfordiffusions(seeforexample,RogersandWilliams(1990),p.338).Remark8.8:ItcanbeshownthatX,Xistheconditionalquadraticvari-ationof[X,X]conditionedonthepredictableeventsP.8.14RandomMeasuresandCanonicalDecom-positionThecanonicaldecompositionofsemimartingaleswithjumpsusestheconceptsofarandommeasureanditscompensator,aswellasintegralswithrespecttorandommeasures.Wedonotusethismaterialelsewhereinthebook.However,thecanonicaldecompositionisoftenmetinresearchpapers.RandomMeasureforaSingleJumpLetξbearandomvariable.ForaBorelsetA⊂IRdefineµ(ω,A)=IA(ξ(ω))=I(ξ(ω)∈A).(8.70)Thenµisarandommeasure,meaningthatforeachω∈Ω,µ(ω,A)isameasurewhenAvaries,A∈B(IR).Its(random)distributionfunctionhasasinglejumpofsize1atξ.ThefollowingrandomStieltjesintegralsconsistofasingleterm:xµ(ω,dx)=ξ(ω),andforafunctionh,h(x)µ(ω,dx)=h(ξ(ω)).IRIR(8.71)Thereisaspecialnotationforthisrandomintegralh∗µ:=h(x)µ(dx).(8.72)IR 8.14.RANDOMMEASURESANDCANONICALDECOMPOSITION245RandomMeasureofJumpsanditsCompensatorinDiscreteTimeLetX0,...Xn,...beasequenceofrandomvariables,adaptedtoFn,andletξn=∆Xn=Xn−Xn−1.Letµn=IA(ξn)bejumpmeasures,andletνn(A)=E(µn(A)|Fn−1)=E(IA(ξn)|Fn−1)=P(ξn∈A|Fn−1)betheconditionaldistributions,n=0,1,....Define
n
nµ((0,n],A)=µi(A),andν((0,n],A)=νi(A).(8.73)i=1i=1ThenforeachAthesequenceµ((0,n],A)−ν((0,n],A)isamartingale.Themeasureµ((0,n],A)iscalledthemeasureofjumpsofthesequenceXn,andν((0,n],A)itscompensator(Adoesnotinclude0).Clearly,themeasureµ={µ((0,n])}n≥1admitsrepresentationµ=ν+(µ−ν),(8.74)whereν={ν((0,n])}n≥1ispredictable,andµ−ν={µ((0,n])−ν((0,n])}n≥1isamartingale.ThisisDoob’sdecompositionforrandommeasures.Withnotation(8.72)Xn=(x∗µ)n.(8.75)Regularconditionaldistributionsexistandforafunctionh(x)theconditionalexpectationscanbewrittenasintegralswithrespecttotheseE(h(ξn)|Fn−1)=h(x)νn(dx),IRprovidedh(ξn)isintegrable,E|h(ξn)|<∞.Assumenowthatξnareintegrable,thenitsDoob’sdecomposition(8.4)
n
nXn=X0+E(ξi|Fi−1)+(ξi−E(ξi|Fi−1))=X0+An+Mn.(8.76)i=1i=1Usingrandommeasuresandtheirintegrals,wecanexpressAnandMnas
n
nAn=E(ξi|Fi−1)=xνi(dx)=(x∗ν)n,(8.77)i=1i=1IR
n
nMn=(ξi−E(ξi|Fi−1))=x(µi(dx)−νi(dx))=(x∗(µ−ν))n.i=1i=1IRThusthesemimartingaledecompositionofXisgivenbyusingtherandommeasureanditscompensatorXn=X0+(x∗ν)n+(x∗(µ−ν))n.(8.78) 246CHAPTER8.CALCULUSFORSEMIMARTINGALESHowever,thejumpsofX,ξn=∆Xn,maynotbeintegrable.Thentheterm(x∗ν)nisnotdefined.Inthiscaseatruncationfunctionisused,suchash(x)=xI(|x|≤1),andasimilardecompositionisachieved,calledthecanonicaldecomposition,
n
n
nXn=X0+E(h(ξi)|Fi−1)+(h(ξi)−E(ξi|Fi−1))+(ξi−h(ξi))i=1i=1i=1=X0+(h∗ν)n+(h∗(µ−ν))n+((x−h(x))∗µ)n.(8.79)Theaboverepresentationhaswell-definedtermsandinadditionithasanotheradvantagethatcarriesovertothecontinuoustimecase.RandomMeasureofJumpsanditsCompensatorLetXbeasemimartingale.Forafixedtconsiderthejump∆X(t).Takingξ=∆X(t),weobtainthemeasureofthejumpattµ({t},A)=IA(∆X(t)),withxµ({t},dx)=∆X(t).(8.80)IR+00NowconsiderthemeasureofjumpsofX(inIR×IR,withIR=IR)
µ((0,t]×A)=IA(∆X(s))(8.81)01(sincex−h(x)=0for|x|≤1).Sincethesumofsquaresofjumpsisfinite,(Corollary8.31,(8.51)),theabovesumhasonlyfinitelymanyterms,henceitisfinite.ThusthefollowingcanonicaldecompositionofasemimartingaleisobtainedcmX(t)=X(0)+A(t)+X(t)+(h(x)∗(µ−ν))(t)+((x−h(x))∗µ)(t),tt=X(0)+A(t)+Xcm(t)+xd(µ−ν)+xdµ,0|x|≤10|x|>1 8.15.EXERCISES247whereAisapredictableprocessoffinitevariation,Xcmisacontinuousmartin-galecomponentofX,µisthemeasureofjumpsofX,andνitscompensator.ForaproofseeLiptserandShiryayev(1989),p.188,Shiryaev(1999),p.663.LetC=Xcm,Xcm(italwaysexistsforcontinuousprocesses).Thefollowingthreeprocessesappearinginthecanonicaldecomposition(A,C,ν)arecalledthetripletofpredictablecharacteristicsofthesemimartingaleX.Notes.MaterialforthischapterisbasedonProtter(1992),RogersandWilliams(1990),Metivier(1982),LiptserandShiryayev(1989),Shiryaev(1999).8.15ExercisesExercise8.1:Letτ1<τ2bestoppingtimes.ShowthatI(τ1,τ2](t)isasimplepredictableprocess.Exercise8.2:LetH(t)bearegularadaptedprocess,notnecessarilyleft-continuous.Showthatforanyδ>0,H(t−δ)ispredictable.Exercise8.3:Showthatacontinuousprocessislocallyintegrable.Showthatacontinuouslocalmartingaleislocallysquareintegrable.TExercise8.4:MisalocalmartingaleandEH2(s)d[M,M](s)<∞.Show0tthatH(s)dM(s)isasquareintegrablemartingale.01Exercise8.5:FindthevarianceofN(t−)dM(t),whereMisthecompen-0satedPoissonprocessM(t)−t.Exercise8.6:IfSandTarestoppingtimes,showthat1.S∧TandS∨Tarestoppingtimes.2.Theevents{S=T},{S≤T}and{St}=B∩{U>t}}isaσ-field.2.ShowthatFtisaright-continuousfiltrationon(Ω,F,P).3.ShowthatUisastoppingtimeforFt. 248CHAPTER8.CALCULUSFORSEMIMARTINGALES4.WhatareF0,FU−andFUequalto?Exercise8.8:LetU1,U2,...be(strictly)positiverandomvariablesonaprobabilityspace(Ω,F,P),andG1,G2,...besub-σ-fieldsofF.Supposethatforalln,U1,U2,...,UnareGn-measurableanddenotebyTntherandomnvariablei=1Ui.SetFt=n{A∈F:∃Bn∈GnsuchthatA∩{Tn>t}=Bn∩{Tn>t}}.1.ShowthatFtisaright-continuousfiltrationon(Ω,F,P).2.Showthatforalln,TnisastoppingtimeforFt.3.SupposethatlimnTn=∞a.s.ShowthatFTn=Gn+1andFTn−=Gn.Exercise8.9:LetB(t)beaBrownianmotionandH(t)beapredictabletprocess.ShowthatM(t)=H(s)dB(s)isaBrownianmotionifandonlyif0Leb({t:|H(t)|=1})=0a.s.Exercise8.10:LetTbeastoppingtime.ShowthattheprocessM(t)=2B(t∧T)−B(t)obtainedbyreflectingB(t)attimeT,isaBrownianmotion.Exercise8.11:LetB(t)andN(t)berespectivelyaBrownianmotionandaPoissonprocessonthesamespace.DenotebyN¯(t)=N(t)−tthecompensatedPoissonprocess.Showthatthefollowingprocessesaremartingales:B(t)N¯(t),E(B)(t)N¯(t)andE(N¯)(t)B(t).Exercise8.12:X(t)solvestheSDEdX(t)=µX(t)dt+aX(t−)dN(t)+σX(t)dB(t).FindtheconditionforX(t)tobeamartingale.Exercise8.13:FindthepredictablerepresentationforY=B5(1).Exercise8.14:FindthepredictablerepresentationforthemartingaleeB(t)−t/2.1Exercise8.15:LetY=sign(B(s))dB(s).ShowthatYhasaNormal0distribution.ShowthatthereisnodeterministicfunctionH(s),suchthat1Y=H(s)dB(s).ThisshowsthattheassumptionthatY,Barejointly0GaussianinTheorem8.35isindispensable.Exercise8.16:Findthequadraticvariationof|B(t)|.Exercise8.17:(StochasticLogarithm)LetUbeasemimartingalesuchthatU(t)andU(t−)areneverzero.ShowthatthereexistsauniquesemimartingaleXwithX(0)=0,(X=L(U))suchdU(t)thatdX(t)=,andU(t−)'')''*'U(t)'1td(s)
'U(s)'U(s)X(t)=ln''''+−ln''''+1−,U(0)20U2(s−)U(s−)U(s−)s≤t(8.82)seeKallsenandShiryaev(2002). Chapter9PureJumpProcessesInthisChapterweconsiderpurejumpprocesses,thatis,processesthatchangeonlybyjumps.CountingprocessesandMarkovJumpprocessesaredefinedandtheirsemimartingalerepresentationisgiven.Thisrepresentationallowsustoseetheprocessasasolutiontoastochasticequationdrivenbydiscontinuousmartingales.9.1DefinitionsAcountingprocessisdeterminedbyasequenceofnon-negativerandomvari-ablesTn,satisfyingTnt}∈Ft).Clearly,Fτ−⊂Fτ.Clearly,thearrivaltimesTnarestoppingtimesforIF.Theyareusuallytakenasalocalizingsequence.NotethatFTn=σ((Ti,Zi),i≤n)andthatX(Tn−)=X(Tn−1),sincefortsatisfyingTn−1≤tt.dM(t)=d(N−A)(t)=dN(t)−dA(t)isthatpartofdN(t)thatcannotbeforeseenfromtheobservationsofNover[0,t).Thenextresultshowstherelationbetweenthesharpbracketofthemartin-galeandthecompensatorinacountingprocess.Sincetheproofisastraightapplicationofstochasticcalculusrules,itisgivenbelow.Itisusefulforcalcu-lationofthevarianceofM,indeedbyTheorem8.24EM2(t)=EM,M(t).tTheorem9.4LetM=N−A.ThenM,M(t)=(1−∆A(s))dA(s).In0particular,ifAiscontinuous,thenM,M(t)=A(t).Proof:Byintegrationbyparts(9.8),wehavet
22M(t)=2M(s−)dM(s)+(∆M(s)).(9.13)0s≤tUse∆M(s)=∆N(s)−∆A(s)andexpandthesumtoobtain



(∆M(s))2=∆N(s)−2∆N(s)∆A(s)+(∆A(s))2,s≤ts≤ts≤ts≤twhereweusedthatsinceNisasimpleprocess(∆N(s))2=∆N(s).Thusweobtainbyusingformula(9.9)andN=M+A,tttM2(t)=2M(s−)dM(s)+N(t)−2∆A(s)dN(s)+∆A(s)dA(s)000tt=2M(s−)+1−2∆A(s)dM(s)+(1−∆A(s))dA(s).00 254CHAPTER9.PUREJUMPPROCESSESTheprocessinthefirstintegralispredictable,becauseM(s−)isadaptedandleft-continuous,A(s)ispredictable,sothat∆A(s)isalsopredictable.There-forethefirstintegralisalocalmartingale.Thesecondintegralispredictable,sinceAispredictable.ThustheaboveequationistheDoob-Meyerdecom-positionofM2.TheresultnowfollowsbytheuniquenessoftheDoob-Meyerdecomposition.
Wegiveexamplesofprocessesandtheircompensatorsnext.PointProcessofaSingleJumpLetTbearandomvariableandtheprocessNhaveasinglejumpattherandomtimet,thatis,N(t)=I(T≤t).LetFbethedistributionfunctionofT.Theorem9.5ThecompensatorA(t)ofN(t)=I(T≤t)isgivenbyt∧TdF(s)A(t)=.(9.14)01−F(s−)Proof:A(t)isclearlypredictable.ToshowthatN(t)−A(t)isamartingale,byTheorem7.17itisenoughtoshowthatEN(S)=EA(S)foranystoppingtimeS.ByTheorem9.1thereexistsanF0-measurable(i.e.almostsurelyconstant)randomvariableζsuchthat{S≥T}={S∧T=T}={ζ∧T=T}={T≤ζ}.ThereforeEN(S)=P(S≥T)=P(T≤ζ)ζζP(T≥t)=dF(t)=dF(t)001−F(t−)ζζ∧TI(T≥t)dF(t)=EdF(t)=E01−F(t−)01−F(t−)S∧TdF(t)=E.01−F(t−)Thus(9.14)isobtained.
t∧TIfFhasadensityf,thenA(t)=h(s)ds,where0f(t)h(t)=,(9.15)1−F(t)iscalledthehazardfunctionanditgivesthelikelihoodoftheoccurrenceofthejumpattgiventhatthejumphasnotoccurredbeforet. 9.4.COUNTINGPROCESSES255CompensatorsofCountingProcessesThenextresultgivesanexplicitformofthecompensatorofageneralcountingprocess.Sincetheproofusesthesameideasasabove,itisomitted.Theorem9.6LetNbeacountingprocessgeneratedbythesequenceTn.De-notebyUn+1=Tn+1−Tntheinter-arrivaltimes,T0=0.LetFn(t)=P(Un+1≤t|T1,...,Tn)denotetheregularconditionaldistributions,andF0(t)=P(T1≤t).ThenthecompensatorA(t)isgivenby
∞t∧Ti+1−t∧TidF(s)iA(t)=.(9.16)01−Fi(s−)i=0NotethatiftheconditionaldistributionsFnintheabovetheoremarecontin-uouswithFn(0)=0,thenbychangingvariableswehaveaadFn(s)dFn(s)==−log(1−Fn(a)),(9.17)01−Fn(s−)01−Fn(s)andequation(9.16)canbesimplifiedaccordingly.RenewalProcessArenewalprocessNisapointprocessinwhichallinter-arrivaltimesareinde-pendentandidenticallydistributed,thatis,T1,T2−T1,...Tn+1−Tnarei.i.d.withdistributionfunctionF(x).InthiscasealltheconditionaldistributionsFnintheprevioustheoremaregivenbyF.AsaresultofTheorem9.6weobtainthefollowingcorollary.Corollary9.7Assumethattheinter-arrivaldistributioniscontinuousandF(0)=0.Thenthecompensatoroftherenewalprocessisgivenby
∞A(t)=−log1−F(t∧Tn−t∧Tn−1).(9.18)n=1StochasticIntensitytDefinition9.8IfA(t)=λ(s)ds,whereλ(t)isapositivepredictablepro-0cess,thenλ(t)iscalledthestochasticintensityofN.NotethatifA(t)isdeterministicanddifferentiablewithderivativeA(t),andtA(t)=A(s)ds,thenλ(t)=A(t)ispredictableandisthestochasticin-0tensity.IfA(t)israndomanddifferentiablewithderivativeA(t),satisfyingttA(t)=A(s)ds,thenλ(t)=A(t−).Indeed,A(t)=A(s−)ds.00 256CHAPTER9.PUREJUMPPROCESSESIfthestochasticintensityexists,thenbydefinitionofthecompensator,tN(t)−λ(s)dsisalocalmartingale.Forcountingprocessesaheuristic0interpretationoftheintensityisgivenbyλ(t)dt=dA(t)=E(dN(t)|Ft−)=P(dN(t)=1|Ft−).(9.19)Ifthestochasticintensityexists,thenthecompensatoriscontinuousandbyTheorem9.4thesharpbracketofthemartingaleM=N−AisgivenbytM,M(t)=λ(s)ds.(9.20)0Example9.2:1.Adeterministicpointprocessisitsowncompensator,soitdoesnothavestochasticintensity.2.Stochasticintensityfortherenewalprocesswithcontinuousinter-arrivaldis-tributionFisgivenbyh(V(t−)),wherehisthehazardfunctionandV(t)=t−TN(t),calledtheageprocess.Thiscanbeseenbydifferentiatingthecom-pensatorin(9.18).3.Stochasticintensityfortherenewalprocesswithadiscreteinter-arrivaldistri-butionFdoesnotexist.Non-homogeneousPoissonProcessesTheorem9.9LetN(t)bepointprocesswithacontinuousdeterministiccom-pensatorA(t).ThenithasindependentPoissondistributedincrements,thatis,thedistributionofN(t)−N(s)isPoissonwithparameterA(t)−A(s),0≤s0,andthesizeofthejumpfromxisintegrablewithmeanm(x).1.ThecompensatorofXisgivenbytA(t)=λ(X(s))m(X(s))ds.(9.31)02.Supposethesecondmomentsofthejumpsarefinite,v(x)=Eξ2(x)<∞.ThenthesharpbracketofthelocalmartingaleM(t)=X(t)−A(t)isgivenbytM,M(t)=λ(X(s))v(X(s))ds.(9.32)0Proof:Theformula(9.31)followsfrom(9.25).Indeed,usingtheexponentialformofFn(9.28),wehave
∞A(t)=m(X(Ti))λ(X(Ti))(t∧Ti+1−t∧Ti).(9.33)i=0NotethatsincetheprocessXhasaconstantvalueX(Ti)onthetimeinterval[Ti,Ti+1),wehaveforafunctionft
∞f(X(s))ds=f(X(Ti))(t∧Ti+1−t∧Ti),(9.34)0i=0andtakingf(x)=λ(x)m(x)givestheresult.Theformula(9.32)followsfromTheorem9.14.
9.6.STOCHASTICEQUATIONFORJUMPPROCESSES2619.6StochasticEquationforJumpProcessesItisclearfromTheorem9.15thataMarkovJumpprocessXhasasemi-martingalerepresentationtX(t)=X(0)+A(t)+M(t)=X(0)+λ(X(s))m(X(s))ds+M(t).(9.35)0ThisistheintegralformofthestochasticdifferentialequationforXdX(t)=λ(X(t))m(X(t))dt+dM(t),(9.36)andisdrivenbythepurelydiscontinuous,finitevariationmartingaleM.Byanalogywithdiffusionstheinfinitesimalmeanisλ(x)m(x)andtheinfinitesimalvarianceisλ(x)v(x).ItisusefultohaveconditionsassuringthatthelocalmartingaleMintherepresentation(9.35)isamartingale.Suchconditionsaregiveninthenextresult.Theorem9.16Supposethatforallxλ(x)E(|ξ(x)|)≤C(1+|x|),(9.37)thenrepresentation(9.35)holdswithazeromeanmartingaleM.Ifinadditionλ(x)v(x)≤C(1+x2),(9.38)thenMissquareintegrablewithtM,M(t)=λ(X(s))v(X(s))ds.(9.39)0Inparticular,wehavethefollowingcorollary.Corollary9.17SupposethattheconditionsoftheaboveTheorem9.16hold.ThentEX(t)=EX(0)+Eλ(X(s))m(X(s))ds,(9.40)0tVar(M(t)−M(0))=Eλ(X(s))v(X(s))ds.(9.41)0TheproofoftheresultcanbefoundinHamzaandKlebaner(1995).AnapplicationofthestochasticequationapproachtothemodelofBirth-DeathprocessesisgiveninChapter13. 262CHAPTER9.PUREJUMPPROCESSESRemark9.2:MarkovJumpprocesseswithcountablyorfinitelymanystatesarecalledMarkovChains.ThejumpvariableshavediscretedistributionsinaMarkovChain,whereastheycanhaveacontinuousdistributioninaMarkovprocess.MarkovJumpprocessesarealsoknownasMarkovChainswithgeneralstatespace.Remark9.3:AmodelforarandomlyevolvingpopulationcanbeservedbyaMarkovChainonnon-negativeintegers.Inthiscasethestatesoftheprocessarethepossiblevaluesforthepopulationsize.ThetraditionalwayofdefiningaMarkovChainisbytheinfinitesimalprobabilities:forintegeriandjandsmallδP(X(t+δ)=j|X(t)=i)=λijδ+o(δ),wherelimδ→0o(δ)/δ=0.Inthissectionwepresentedanalmostsure,pathbypathrepresentationofaMarkovJumpprocess.AnotherrepresentationrelatedtothePoissonprocesscanbefoundinEthierandKurtz(1986).GeneratorsandDynkin’sFormulaLetXbetheMarkovJumpprocessdescribedby(9.27).Supposethatλ(x)isbounded,thatissupxλ(x)<∞.DefinethefollowinglinearoperatorLactingonboundedfunctionsbyLf(x)=λ(x)Ef(x+ξ(x))−f(x)=λ(x)mf(x).(9.42)Theorem9.18(Dynkin’sFormula)LetLbeasabove,anddefineMf(t)bytMf(t)=f(X(t))−f(X(0))−Lf(X(s))ds.(9.43)0ThenMfisamartingale,andconsequentlytEf(X(t))=f(X(0))+ELf(X(s))ds.(9.44)0Proof:TheproofoftheseformulaefollowsfromTheorem9.13andTheo-rem9.15.Theprocessf(X(t))isapurejumpprocesswiththesamearrivaltimesT,andjumpsofsizeZ=f(X(T))−f(X(T−))(inthiscaseitisnnnnalsoMarkov).Thejumpsarebounded,sincefisbounded.ThemeanofthejumpfromxisgivenbyE(Z|F,X(T)=x)=Ef(x+ξ(x))−n+1Tnnf(x)=mf(x).Thereforethecompensatoroff(X(t))is,fromTheorem9.15,tλ(X(s))m(X(s))ds.ThisimpliesthatMfisalocalmartingale.Ifλ(x)is0fbounded,thenM(t)isboundedbyaconstantCtonanyfinitetimeinterval[0,t].Sinceaboundedlocalmartingaleisamartingale,Mfisamartingale.Moreoveritissquareintegrableonanyfinitetimeinterval.
9.7.EXPLOSIONSINMARKOVJUMPPROCESSES263ThelinearoperatorLin(9.42)iscalledthegeneratorofX.Itcanbeshownthatwithitshelponecandefineprobabilitiesonthespaceofofright-continuousfunctions,sothatthecoordinateprocessistheMarkovprocess,cf.solutiontothemartingaleproblemfordiffusions.Theorem9.16abovecanbeseenasageneralizationofDynkin’sformulafortheidentityfunctionf(x)=xandthequadraticfunctionf(x)=x2.TheseareparticularcasesofamoregeneraltheoremTheorem9.19Assumethattherearenoexplosions,thatis,Tn↑∞.Sup-posethatforanunboundedfunctionfthereisaconstantCsuchthatforallxλ(x)E|f(x+ξ(x))−f(x)|≤C(1+|f(x)|).(9.45)SupposealsothatE|f(X(0))|<∞.Thenforallt,0≤t<∞,E|f(X(t))|<∞,moreoverMfin(9.43)isamartingaleand(9.44)holds.FortheproofseeHamzaandKlebaner(1995).9.7ExplosionsinMarkovJumpProcessesLetX(t)beajumpMarkovprocesstakingvaluesinIR.Iftheprocessisinstatesxthenitstaysthereforanexponentiallengthoftimewithmeanλ−1(x)afterwhichitjumpsfromx.Ifλ(x)→∞forasetofvaluesx,thentheexpecteddurationofstayinstatex,λ−1(x)→0,andthetimespentinxbecomesshorterandshorter.Itcanhappenthattheprocessjumpsinfinitelymanytimesinafinitetimeinterval,thatis,limn→∞Tn=T∞<∞.Thisphenomenoniscalledexplosion.Theterminologycomesfromthecasewhentheprocesstakesonlyintegervaluesandλ(x)cantendtoinfinityonlywhenx→∞.Inthiscasethereareinfinitelymanyjumpsonlyiftheprocessreachesinfinityinfinitetime.Whentheprocessdoesnotexplodeitiscalledregular.Inotherwords,theregularityassumptionisT∞=limTn=∞.a.s.(9.46)n→∞IfP(T∞<∞)>0,thenitsaidthattheprocessexplodesontheset{T∞<∞}.Anecessaryandsufficientconditionfornon-explosionisgivenbyTheorem9.20
∞1=∞.a.s.(9.47)λ(X(Tn)0 264CHAPTER9.PUREJUMPPROCESSESProof:Wesketchtheproof,seeBreiman(1968)fordetails.Letνnbeasequenceofindependentexponentiallydistributedrandomvariableswithparameterλn.Thenitiseasytosee(bytakingaLaplacetransform)thatn∞10νi<∞convergesindistributionifandonlyif0λn<∞.Usingtheresultthataseriesofindependentrandomvariablesconvergesalmostsurelyif∞andonlyifitconvergesindistribution,wehave0νn<∞almostsurelyifandonlyif∞1<∞.Condition(9.47)nowfollows,sincetheconditional0λndistributionofTn+1−Tn,giventheinformationuptothen-thjump,FTn,isexponentialwithparameterλ(X(Tn)).
Condition(9.47)ishardtocheckingeneral,sinceitinvolvesthevari-ablesX(Tn).Asimplecondition,whichisessentiallyintermsofthefunctionλ(x)m(x)(thedrift),isgivenbelow.Theorem9.21AssumethatX≥0,andthereexistsamonotonefunction∞f(x)suchthatλ(x)m(x)≤f(x)anddx=∞.ThentheprocessXdoes0f(x)notexplode,thatis,P(X(t)<∞forallt,0≤t<∞)=1.Accordingtothis,thefirstconditionoftheresultonintegrability,Theorem9.16,guaranteesthattheprocessdoesnotexplode.ProofofTheorem9.21andothersharpsufficientconditionsforregularityintermsoftheparametersoftheprocessaregiveninKerstingandKlebaner(1995),(1996).Notes.MaterialforthischapterisbasedonLiptserandShiryayev(1974),(1989),Karr(1986),andtheresearchpapersquotedinthetext. 9.8.EXERCISES2659.8ExercisesxdF(s)Exercise9.1:LetU≥0anddenoteh(x)=,whereF(x)isthe01−F(s−)adistributionfunctionofU.Showthatfora>0Eh(U∧a)=dF(x).0Exercise9.2:Showthatwhenthedistributionofinter-arrivaltimesisexpo-nentialtheformula(9.18)givesthecompensatorλt,henceN(t)isaPoissonprocess.Exercise9.3:Showthatwhenthedistributionofinter-arrivaltimesisGeo-metricthenN(t)hasaBinomialdistribution.Exercise9.4:Prove(9.25).Exercise9.5:LetX(t)beapurejumpprocesswithfirstkmoments.Letm(x)=Eξk(x)andassumeforalli=1,...,k,λ(x)E|ξ(x)|k≤C|x|i.Showkthatk
−1)k*tEXk(t)=EXk(0)+Eλ(X(s))Xi(s)m(X(s))dsk−ii0i=0 Thispageintentionallyleftblank Chapter10ChangeofProbabilityMeasureInthischapterwedescribewhathappenstorandomvariablesandprocesseswhentheoriginalprobabilitymeasureischangedtoanequivalentone.ChangeofmeasureforprocessesisdonebyusingGirsanov’stheorem.10.1ChangeofMeasureforRandomVariablesChangeofMeasureonaDiscreteProbabilitySpaceWestartwithasimpleexample.LetΩ={ω1,ω2}withprobabilitymeasurePgivenbyp(ω1)=p,p(ω2)=1−p.Definition10.1QisequivalenttoP(Q∼P)iftheyhavesamenullsets,ie.Q(A)=0ifandonlyifP(A)=0.LetQbeanewprobabilitymeasureequivalenttoP.Inthisexample,thismeansthatQ(ω1)>0andQ(ω2)>0(or00,suchthatEP(Λ)=1,anddefineQby(10.1).ThenQisaprobability,becauseQ(ωi)=Λ(ωi)P(ωi)>0,i=1,2,andQ(Ω)=Q(ω1)+Q(ω2)=Λ(ω1)P(ω1)+Λ(ω2)P(ω2)=EP(Λ)=1.QisequivalenttoP,sinceΛisstrictlypositive.ThuswehaveshownthatforanyequivalentchangeofmeasureQ∼PthereisapositiverandomvariableΛ,suchthatEP(Λ)=1,andQ(ω)=Λ(ω)P(ω).Thisisthesimplestversionofthegeneralresult,theRadon-NikodymTheorem.TheexpectationunderQofarandomvariableXisgivenby,EQ(X)=EP(ΛX).(10.4)BytakingindicatorsI(X∈A)weobtainthedistributionofXunderQQ(X∈A)=EP(ΛI(X∈A)).Theseformulae,obtainedhereforasimpleexample,holdalsoingeneral.ChangeofMeasureforNormalRandomVariablesConsiderfirstachangeofaNormalprobabilitymeasureonIR.Letµbeanyrealnumber,fµ(x)denotetheprobabilitydensityofN(µ,1)distribution,andPµtheN(µ,1)probabilitymeasureonIR(B(IR)).Then,itiseasytoseethat1−1(x−µ)2µx−µ2/2fµ(x)=√e2=f0(x)e.(10.5)2πPut2µx−µ/2Λ(x)=e.(10.6)Thentheaboveequationreadsfµ(x)=f0(x)Λ(x).(10.7) 10.1.CHANGEOFMEASUREFORRANDOMVARIABLES269Bythedefinitionofadensityfunction,theprobabilityofasetAontheline,istheintegralofthedensityoverthisset,P(A)=f(x)dx=dP.(10.8)AAIninfinitesimalnotationsthisrelationiswrittenasdP=P(dx)=f(x)dx.(10.9)Hencetherelationbetweenthedensities(10.7)canbewrittenasarelationbetweenthecorrespondingprobabilitymeasuresfµ(x)dx=f0(x)Λ(x)dx,andPµ(dx)=Λ(x)P0(dx).(10.10)Byapropertyoftheexpectation(integral)(2.3),ifarandomvariableX≥0thenEX=0ifandonlyifP(X=0)=1,Pµ(A)=Λ(x)P0(dx)=EP0(IAΛ)=0AimpliesP0(IAΛ=0)=1.SinceΛ(x)>0forallx,thisimpliesP0(A)=0.UsingthatΛ<∞,theotherdirectionfollows:ifP0(A)=0thenPµ(A)=EP0(IAΛ)=0,whichprovesthatthesemeasureshavesamenullsetsandareequivalent.AnothernotationforΛisdPµdPµµx−µ2/2Λ=,(x)=e.(10.11)dP0dP0ThisshowsthatanyN(µ,1)probabilityisobtainedbyanequivalentchangeofprobabilitymeasurefromtheN(0,1)distribution.Wenowgivethesameresultintermsofchangingthedistributionofarandomvariable.2Theorem10.2LetXhaveN(0,1)underP,andΛ(X)=eµX−µ/2.DefinemeasureQbydQQ(A)=Λ(X)dP=EP(IAΛ(X))or(X)=Λ(X).(10.12)AdPThenQisanequivalentprobabilitymeasure,andXhasN(µ,1)distributionunderQ. 270CHAPTER10.CHANGEOFPROBABILITYMEASUREProof:WeshowfirstthatQdefinedby(10.12)isindeedaprobabil-22ity.Q(Ω)=E(1)=E(1Λ(X))=E(eµX−µ/2)=e−µ/2E(eµX)=1.QPPP2ThelastequalityisbecauseE(eµX)=eµ/2fortheN(0,1)randomvari-Pable.OtherpropertiesofprobabilityQfollowfromthecorrespondingprop-ertiesoftheintegral.ConsiderthemomentgeneratingfunctionofXunder22Q.E(euX)=E(euXΛ(X))=E(e(u+µ)X−µ/2)=e−µ/2E(e(u+µ)X)=QPPP2euµ+u/2,whichcorrespondstotheN(µ,1)distribution.
Remark10.1:IfPisaprobabilitysuchthatXisN(0,1),thenX+µhasN(µ,1)distributionunderthesameP.Thisisanoperationontheoutcomesx.WhenwechangetheprobabilitymeasurePtoQ=Pµ,weleavetheoutcomesastheyare,butassigndifferentprobabilitiestothem(morepreciselytosetsofoutcomes).UnderthenewmeasurethesameXhasN(µ,1)distribution.Example10.1:(Simulationsofrareevents)Changeofprobabilitymeasureisusefulinsimulationandestimationofprobabil-itiesofrareevents.ConsiderestimationofPr(N(6,1)<0)bysimulations.This−10probabilityisabout10.Directsimulationisdonebytheexpression
n1Pr(N(6,1)<0)≈I(xi<0),(10.13)ni=1wherexi’saretheobservedvaluesfromtheN(6,1)distribution.Notethatina6millionruns,n=10,weshouldexpectnovaluesbelowzero,andtheestimateis0.LetQbetheN(6,1)distributiononIR.ConsiderQaschangedfromN(0,1)=P,asfollowsdQdN(6,1)µx−µ2/26x−18(x)=(x)=Λ(x)=e=e.dPdN(0,1)Sowehave6x−18Q(A)=N(6,1)(A)=EPΛ(x)I(A)=EN(0,1)eI(A).InourcaseA=(−∞,0),andPr(N(6,1)<0)=Q(A).Thusweareledtothefollowingestimate
n
n16xi−18−1216(xi−1)Pr(N(6,1)<0)=Q(A)≈eI(xi<0)=eeI(xi<0),nni=1i=1(10.14)withxigeneratedfromN(0,1)distribution.Ofcourse,abouthalfoftheobservationsxi’swillbenegative,resultinginamorepreciseestimate,evenforasmallnumberofrunsn.Nextwegivetheresultforrandomvariables,setupinawaysimilartoGir-sanov’stheoremforprocesses,changingmeasuretoremovethemean. 10.2.CHANGEOFMEASUREONAGENERALSPACE271Theorem10.3(Removalofthemean)LetXhaveN(0,1)distributionunderP,andY=X+µ.ThenthereisanequivalentprobabilityQ∼P,2suchthatYhasN(0,1)underQ.(dQ/dP)(X)=Λ(X)=e−µX−µ/2.Proof:Similarlytothepreviousproof,Qisaprobabilitymeasure,withthesamenullsetsasP.uYu(X+µ)(u−µ)X+uµ−1µ2uµ−1µ2(u−µ)XEQ(e)=EP(eΛ(X))=EP(e2)=e2EP(e).2UsingthemomentgeneratingfunctionofN(0,1),E(e(u−µ)X)=e(u−µ)/2,P2whichgivesE(euY)=eu/2,establishingY∼N(0,1)underQ.Q
Finally,wegivethechangeofoneNormalprobabilitytoanother.Theorem10.4LetXhaveN(µ,σ2)distribution,callitP.DefineQby11(X−µ1)2(X−µ2)2σ12σ2−2σ2(dQ/dP)(X)=Λ(X)=e12.(10.15)σ2ThenXhasN(µ,σ2)distributionunderQ.22Proof:TheformofΛiseasilyverifiedastheratioofNormaldensities.ThisprovesthestatementforPandQonIR.Onageneralspace,theproofissimilartotheabove,byworkingoutthemomentgeneratingfunctionofXunderQ.
10.2ChangeofMeasureonaGeneralSpaceLettwoprobabilitymeasuresPandQbedefinedonthesamespace.Definition10.5QiscalledabsolutelycontinuouswithrespecttoP,writtenasQP,ifQ(A)=0wheneverP(A)=0.PandQarecalledequivalentifQPandPQ.Theorem10.6(Radon-Nikodym)IfQP,thenthereexistsarandomvariableΛ,suchthatΛ≥0,EPΛ=1,andQ(A)=EP(ΛI(A))=ΛdP(10.16)AforanymeasurablesetA.ΛisP-almostsurelyunique.Conversely,ifthereex-istsarandomvariableΛwiththeabovepropertiesandQisdefinedby(10.16),thenitisaprobabilitymeasureandQP. 272CHAPTER10.CHANGEOFPROBABILITYMEASURETherandomvariableΛintheabovetheoremiscalledtheRadon-NikodymderivativeorthedensityofQwithrespecttoP,andisdenotedbydQ/dP.Itfollowsfrom(10.16)thatifQP,thenexpectationsunderPandQarerelatedbyEQX=EP(ΛX),(10.17)foranyrandomvariableXintegrablewithrespecttoQ.Calculationsofexpectationsaresometimesmadeeasierbyusingachangeofmeasure.XExample10.2:(ALognormalcalculationforafinancialoption)EeI(X>a),XXX−µ−1whereXhasN(µ,1)distribution(P).TakeΛ(X)=e/Ee=e2,anddQ=Λ(X)dP.ThenEΛ(X)=1,0<Λ(X)<∞,soQ∼P.1Xµ+EPeI(X>a)=e2EPΛ(X)I(X>a)=EQI(X>a)=Q(X>a).uXuX(u+1)X−µ−1uµ+u+u2/2EQ(e)=EP(eΛ(X))=EP(e2)=e,whichisthetransformofN(µ+1,1)distribution.ThustheQdistributionofXisXN(µ+1,1),andE(eI(X>a))=Q(X>a)=Pr(N(µ+1,1)>a)=1−Φ(a−µ−1).Wegivethedefinitionofthe“opposite”concepttotheabsolutecontinuityoftwomeasures.Definition10.7TwoprobabilitymeasuresPandQdefinedonthesamespacearecalledsingularifthereexistasetAsuchthatP(A)=0andQ(A)=1.Singularitymeansthatbyobservinganoutcomewecandecidewithcertaintyontheprobabilitymodel.+Example10.3:1.LetΩ=IR,Pistheexponentialdistributionwithparameter1,andQisthePoissondistributionwithparameter1.ThenPandQaresingular.Indeedthesetofnon-negativeintegershasQprobability1,andPprobability0.2.WeshallseelaterthattheprobabilitymeasuresinducedbyprocessesσB(t),whereBisaBrownianmotion,aresingularfordifferentvaluesofσ.Expectationsforanabsolutelycontinuouschangeofprobabilitymeasurearerelatedbytheformula(10.17).Conditionalexpectationsarerelatedbyanequation,similartoBayesformula.Theorem10.8(GeneralBayesformula)LetGbeasub-σ-fieldofFonwhichtwoprobabilitymeasuresQandParedefined.IfQPwithdQ=ΛdPandXisQ-integrable,thenΛXisP-integrableandQ-a.s.EP(XΛ|G)EQ(X|G)=.(10.18)EP(Λ|G) 10.2.CHANGEOFMEASUREONAGENERALSPACE273Proof:WecheckthedefinitionofconditionalexpectationgivenG,(2.15)foranyboundedG-measurablerandomvariableξEQ(ξX)=EQ(ξEQ(X|G)).Clearlytherhsof(10.18)isG-measurable.)*)*EP(XΛ|G)EP(XΛ|G)EQξ=EPΛξEP(Λ|G)EP(Λ|G))*EP(XΛ|G)=EPEP(Λ|G)ξEP(Λ|G)=EP(EP(XΛ|G)ξ)=EP(XΛξ)=EQ(Xξ).
Intherestofthissectionweaddressthegeneralcaseoftwomeasures,notnecessarilyequivalent.LetPandQbetwomeasuresonthesameprobabilityP+Qspace.Considerthemeasureν=.Then,clearly,PνandQν.2Therefore,bytheRadon-NikodymTheorem,thereexistΛ1andΛ2,suchthatdP/dν=ΛanddQ/dν=Λ.Introducethenotationx(+)=1/x,ifx=0,12and0otherwise,sothatx(+)x=I(x=0).Wehave(+)(+)Q(A)=Λ2dν=Λ2Λ1Λ1+1−Λ1Λ1dνAA(+)(+)=Λ2Λ1Λ1dν+(1−Λ1Λ1)Λ2dνAA(+)(+)=Λ2Λ1dP+(1−Λ1Λ1)dQAAcs:=Q(A)+Q(A),c(+)whereQ(A)=AΛ2Λ1dPisabsolutelycontinuouswithrespecttoP,ands(+)Q(A)=A(1−Λ1Λ1)dQ.ThesetA={Λ1>0}hasPprobabilityone,P(A)=Λ1dν=Λ1dν=P(Ω)=1,AΩs(+)sbuthasmeasurezerounderQ,becauseonthissetΛ1Λ1=1andQ(A)=0.sThisshowsthatQissingularwithrespecttoP.ThuswehaveTheorem10.9(LebesgueDecomposition)LetPandQbetwomeasurescsconthesameprobabilityspace.ThenQ(A)=Q(A)+Q(A),whereQPsandQ⊥P. 274CHAPTER10.CHANGEOFPROBABILITYMEASURE10.3ChangeofMeasureforProcessesSincerealizationsofaBrownianmotionarecontinuousfunctions,theprobabil-ityspaceistakentobethesetofcontinuousfunctionson[0,T],Ω=C([0,T]).Becausewehavetodescribethecollectionofsetstowhichtheprobabilityisassigned,theconceptsofanopen(closed)setisneeded.Thesearedefinedintheusualwaywiththehelpofthethedistancebetweentwofunctionstakenassupt≤T|w1(t)−w2(t)|.Theσ-fieldistheonegeneratedbyopensets.Probabil-itymeasuresaredefinedonthemeasurablesubsetsofΩ.Ifω=w[0,T]denotesacontinuousfunction,thenthereisaprobabilitymeasurePonthisspacesuchthatthe“coordinate”processB(t,ω)=B(t,w[0,T])=w(t)isaBrownianmo-tion,PistheWienermeasure,seeSection5.7.NotethatalthoughaBrownianmotioncanbedefinedon[0,∞),theequivalentchangeofmeasureisdefinedonlyonfiniteintervals[0,T].ArandomvariableonthisspaceisafunctionX:Ω→IR,andX(ω)=X(w[0,T]),alsoknownasafunctionalofBrown-ianmotion.Sinceprobabilitiescanbeobtainedasexpectationofindicators,P(A)=EIA,itisimportanttoknowhowtocalculateexpectations.E(X)isgivenasanintegralwithrespecttotheWienermeasureE(X)=X(w[0,T])dP.(10.19)ΩInparticular,ifX(w[0,T])=h(w(T)),forsomefunctionofrealargumenth,thenE(X)=h(w(T))dP=E(h(B(T)),(10.20)ΩwhichcanbeevaluatedbyusingtheN(0,T)distributionofB(T).Similarly,ifXafunctionoffinitelymanyvaluesofw,X(w[0,T])=h(w(t1),...,w(tn)),thentheexpectationcanbecalculatedbyusingthemultivariateNormaldis-tributionofB(t1),...,B(tn).However,forafunctionalwhichdependsonthewholepathw[0,T],suchasintegralsofw(t)ormaxt≤Tw(t),thedistributionofthefunctionalisrequired(10.21).ItcanbecalculatedbyusingthedistributionofX,FX,asforanyrandomvariableE(X)=X(w[0,T])dP=xdFX(x).(10.21)ΩIRIfPandQareequivalent,thatis,theyhavesamenullsets,thenthereexistsarandomvariableΛ(theRadon-NikodymderivativeΛ=dQ/dP),suchthattheprobabilitiesunderQaregivenbyQ(A)=ΛdP.Girsanov’stheoremAgivestheformofΛ.Firstwestateageneralresult,thatfollowsdirectlyfromTheorem10.8,forthecalculationofexpectationsandconditionalexpectationsunderanab-solutelycontinuouschangeofmeasure.ItisalsoknownasthegeneralBayesformula. 10.3.CHANGEOFMEASUREFORPROCESSES275Theorem10.10LetΛ(t)beapositiveP-martingalesuchthatEP(Λ(T))=1.DefinethenewprobabilitymeasureQbytherelationQ(A)=Λ(T)dP,(soAthatdQ/dP=Λ(T)).ThenQisabsolutelycontinuouswithrespecttoPandforaQ-integrablerandomvariableXEQ(X)=EPΛ(T)X.(10.22))*Λ(T)EQ(X|Ft)=EPX|Ft,(10.23)Λ(t)andifXisFtmeasurable,thenfors≤t)*Λ(t)EQ(X|Fs)=EPX|Fs.(10.24)Λ(s)Λ(T)Proof:Itremainstoshow(10.24).EQ(X|Fs)=EPΛ(s)X|Fs=EE(Λ(T)X|F)|F=EX1E(Λ(T)|F)|F=EΛ(t)X|F.PPΛ(s)tsPΛ(s)PtsPΛ(s)s
Thefollowingresultfollowsimmediatelyfrom(10.24).Corollary10.11AprocessM(t)isaQ-martingaleifandonlyifΛ(t)M(t)isaP-martingale.BytakingM(t)=1weobtainaresultisusedinfinancialapplications.Theorem10.12LetΛ(t)beapositiveP-martingalesuchthatEP(Λ(T))=1,anddQ/dP=Λ(T).Then1/Λ(t)isaQ-martingale.Weshownextthatconvergenceinprobabilityispreservedunderanabsolutelycontinuouschangeofmeasure.Theorem10.13LetXn→XinprobabilityPandQP.ThenXn→XinprobabilityQ.Proof:DenoteAn={|Xn−X|>ε}.ThenconvergenceinprobabilityofXntoXmeansP(An)→0.ButQ(An)=EP(ΛIAn).SinceΛisP-integrable,theresultfollowsbydominatedconvergence.
Corollary10.14Thequadraticvariationofaprocessdoesnotchangeunderanabsolutelycontinuouschangeoftheprobabilitymeasure.n2Proof:Thesumsi=0(X(ti+1)−X(ti))approximatingthequadraticvariationconvergeinPprobabilityto[X,X](t).BytheaboveresulttheyconvergetothesamelimitunderanequivalenttoPprobabilityQ.
276CHAPTER10.CHANGEOFPROBABILITYMEASURETheorem10.15(Girsanov’stheoremforBrownianmotion)LetB(t),0≤t≤T,beaBrownianmotionunderprobabilitymeasureP.ConsidertheprocessW(t)=B(t)+µt.DefinethemeasureQbydQ−µB(T)−1µ2TΛ=(B[0,T])=e2,(10.25)dPwhereB[0,T]denotesapathofBrownianmotionon[0,T].ThenQisequivalenttoP,andW(t)isaQ-Brownianmotion.dP1µW(T)−1µ2T(W[0,T])==e2.(10.26)dQΛProof:TheproofusesLevy’scharacterizationofBrownianmotion,asacontinuousmartingalewithquadraticvariationprocesst.QuadraticvariationisthesameunderPandQbyTheorem10.13.Therefore(withaslightabuseofnotation)usingthefactthatµtissmoothanddoesnotcontributetothequadraticvariation,[W,W](t)=[B(t)+µt,B(t)+µt]=[B,B](t)=t.ItremainstoestablishthatW(t)isaQ-martingale.LetΛ(t)=EP(Λ|Ft).BytheCorollary(10.11)toTheorem10.10itisenoughtoshowthatΛ(t)W(t)isaP-martingale.Thisisdonebydirectcalculations.−µB(t)−1µ2tEP(W(t)Λ(t)|Fs)=EP(B(t)+µt)e2|Fs=W(s)Λ(s).
tT2ItturnsoutthatadriftoftheformH(s)dswithH(s)ds<∞can00beremovedsimilarlybyachangeofmeasure.Theorem10.16(Girsanov’stheoremforremovalofdrift)LetB(t)betaP-Brownianmotion,andH(t)issuchthatX(t)=−H(s)dB(s)isde-0fined,moreoverE(X)isamartingale.DefineanequivalentmeasureQbydQT1T2−H(s)dB(s)−H(s)dsΛ=(B)=e020=E(X)(T).(10.27)dPThentheprocesstW(t)=B(t)+H(s)dsisaQ-Brownianmotion.(10.28)0Proof:Theproofissimilartothepreviousoneandweonlysketchit.tWeshowthatW(t)=B(t)+H(s)dsisacontinuousQ-martingalewith0quadraticvariationt.TheresultthenfollowsbyLevy’scharacterization.The 10.3.CHANGEOFMEASUREFORPROCESSES277tquadraticvariationofW(t)=B(t)+H(s)dsist,sincetheintegralis0continuousandisoffinitevariation.W(t)isclearlycontinuous.ToestablishthatW(t)isaQ-martingale,byCorollary10.11,itisenoughtoshowthatΛ(t)W(t)isaP-martingale,withΛ(t)=EP(Λ|Ft).TothisendconsiderΛ(t)E(W)(t)=E(X)tE(W)t=E(X+W+[X,W])t=E(X+B)t,wherewehaveusedtherulefortheproductofsemimartingaleexponentialst(Theorem8.13),[X,W](t)=−H(s)dsandW(t)=B(t)+[X,W](t).Since0XandBarebothP-martingales,soisX+B.ConsequentlyE(X+B)isalsoaP-martingale.HenceΛ(t)E(W)tisaP-martingale,andconsequentlyE(W)tisaQ-martingale.ThisimpliesthatW(t)isalsoaQ-martingale.
Girsanov’stheoremholdsinndimensions.Theorem10.17LetBbeaPn-dimensionalBrownianmotionandW=(W1(t),...,Wn(t)),wheretWi(t)=Bi(t)+Hi(s)ds,0withH(t)=(H1(t),H2(t),...,Hn(t))aregularadaptedprocesssatisfyingT|H(s)|2ds<∞.Let0
ntX(t)=−H·B:=−Hi(s)dBi(s),i=10andassumethatE(X)Tisamartingale.ThenthereisanequivalentprobabilitymeasureQ,suchthatWisaQ-Brownianmotion.QisdeterminedbydQ(B[0,T])=Λ(B[0,T])=E(X)(T).dPCommentthatasufficientconditionforE(X)tobeamartingaleisThe-1T2H(s)dsorem8.17,E(e20)<∞,andforE(H·B)tobeamartingaleis1T2|H(s)|dsEe20<∞.Theproofforndimensionsissimilartoonedimension,usingcalculationsfromExercise10.5.WegiveaversionofGirsanov’stheoremformartingales.Theproofissimilartotheoneabove,andisnotgiven.Theorem10.18LetM1(t),0≤t≤TbeacontinuousP-martingale.LetX(t)beacontinuousP-martingalesuchthatE(X)isamartingale.DefineanewprobabilitymeasureQbydQX(T)−1[X,X](T)=Λ=E(X)(T)=e2.(10.29)dP 278CHAPTER10.CHANGEOFPROBABILITYMEASUREThenM2(t)=M1(t)−[M,X](t)(10.30)isacontinuousmartingaleunderQ.AsufficientconditionforE(X)tobeamartingaleisNovikov’scondition1[X,X]TEe2<∞,orKazamaki’scondition(8.40).ChangeofDriftinDiffusionsLetX(t)beadiffusion,sothatwithaP-BrownianmotionB(t),X(t)satisfiesthefollowingstochasticdifferentialequationwithσ(x,t)>0,dX(t)=µ1(X(t),t)dt+σ(X(t),t)dB(t).(10.31)Letµ1(X(t),t)−µ2(X(t),t)H(t)=,(10.32)σ(X(t),t)anddefineQbydQ=ΛdPwith)·*TTdQ−H(t)dB(t)−1H2(t)dtΛ==E−H(t)dB(t)(T)=e020.(10.33)dP0ByGirsanov’stheorem,providedtheprocessE(H·B)isamartingale,thetprocessW(t)=B(t)+H(s)dsisaQ-Brownianmotion.But0µ1(X(t),t)−µ2(X(t),t)dW(t)=dB(t)+H(t)dt=dB(t)+dt.(10.34)σ(X(t),t)Rearranging,weobtaintheequationforX(t)dX(t)=µ2(X(t),t)dt+σ(X(t),t)dW(t),(10.35)withaQ-BrownianmotionW(t).ThusthechangeofmeasureforaBrow-nianmotiongivenaboveresultsinthechangeofthedriftinthestochasticdifferentialequation.Example10.4:(MaximumofarithmeticBrownianmotion)∗LetW(t)=µt+B(t),whereB(t)isP-Brownianmotion,andW(t)=maxs≤tW(s).Whenµ=0thedistributionofthemaximumaswellasthejointdistributionareknown,Theorem3.21.Wefindthesedistributionswhenµ=0.LetB(t)beaQ-Brownianmotion,thenW(t)isaP-Brownianmotionwith∗Λ=dP/dQisgivenby(10.26).LetA={W(T)∈I1,W(T)∈I2},whereI1,I2areintervalsontheline.Thenweobtainfrom(10.26)∗µW(T)−1µ2TµW(T)−1µ2TP((W(T),W(T))∈A)=e2dQ=EQ(e2I(A)).(10.36)A 10.4.CHANGEOFWIENERMEASURE279∗DenotebyqW,W∗(x,y)andpW,W∗(x,y)thejointdensityof(W(T),W(T))underQandPrespectively.Thenitfollowsfrom(10.36)µx−1µ2TpW,W∗(x,y)dxdy=qW,W∗(x,y)e2dxdy.(10.37){x∈I1,y∈I2}{x∈I1,y∈I2}Thusµx−1µ2TpW,W∗(x,y)=e2qW,W∗(x,y).(10.38)ThedensityqW,W∗isgivenby(3.16).ThejointdensitypW,W∗(x,y)canbecomputed(seeExercise10.6),andthedistributionofthemaximumisfoundbypW,W∗(x,y)dx.10.4ChangeofWienerMeasureGirsanov’sTheorem10.16statesthatiftheWienermeasurePischangedttoQbydQ/dP=Λ=E(H(s)dB(s))whereB(t)isaP-Brownianmo-0tion,andH(t)issomepredictableprocess,thenQisequivalenttoP,andtB(t)=W(t)+H(s)dsforaQ-BrownianmotionW.Inthissectionweprove0theconversethattheRadon-NikodymderivativeofanymeasureQ,equiva-tlenttotheWienermeasureP,isastochasticexponentialofq(s)dB(s)for0somepredictableprocessq.UsingthepredictablerepresentationpropertyofBrownianmartingalesweprovethefollowingresultfirst.Theorem10.19LetIFbeBrownianmotionfiltrationandYbeapositiverandomvariable.IfEY<∞thenthereexistsapredictableprocessq(t)suchT1T2q(t)dB(t)−q(t)dtthatY=(EY)e020.Proof:LetM(t)=E(Y|Ft).ThenM(t),0≤t≤Tisapositiveuniformlytintegrablemartingale.ByTheorem8.35M(t)=EY+H(s)dB(s).Define0q(t)=H(t)/M(t).ThenwehavedM(t)=H(t)dB(t)=M(t)q(t)dB(t)=M(t)dX(t),(10.39)twithX(t)=q(s)dB(s).ThusMisasemimartingaleexponentialofX,0X(t)−1[X,X](t)M(t)=M(0)e2andtheresultfollows.
Itremainstoshowthatqisproperlydefined.Weknowthatforeacht,M(t)>0withprobabilityone.ButtheremaybeanexceptionalsetNt,ofprobabilityzero,onwhichM(t)=0.Asthereareuncountablymanyt’stheunionofNt’smayhaveapositiveprobability(evenprobabilityone),precludingqbeingfinitewithapositiveprobability.Thenextresultshowsthatthisisimpossible.Theorem10.20LetM(t),0≤t≤T,beamartingale,suchthatforanyt,P(M(t)>0)=1.ThenM(t)neverhitszeroon[0,T]. 280CHAPTER10.CHANGEOFPROBABILITYMEASUREProof:Letτ=inf{t:M(t)=0}.UsingtheBasicStoppingequationEM(τ∧T)=EM(T).SowehaveEM(T)=EM(τ∧T)=EM(T)I(τ>T)+EM(τ)I(τ≤T)=EM(T)I(τ>T).ItfollowsthatEM(T)(1−I(τ>T))=0.Sincetherandomvariablesundertheexpectationarenon-negative,thisimpliesM(T)(1−I(τ>T))=0a.s.ThusI(τ>T)=1a.s.ThusthereisanullsetNsuchthatτ>ToutsideN,orP(M(t)>0,forallt≤T)=1.SinceTwasarbitrary,theargumentimpliesthatτ=∞andzeroisneverhit.
Remark10.2:Notethatifforsomestoppingtimeτanon-negativemartin-galeiszero,M(τ)=0,andoptionalstoppingholds,forallt>τE(M(t)|Fτ)=M(τ)=0,thenM(t)=0a.s.forallt>τ.Corollary10.21LetPbetheWienermeasure,B(t)beaP-BrownianmotionandQbeequivalenttoP.Thenthereexistsapredictableprocessq(t),suchthatdQT1T2q(t)dB(t)−q(t)dtΛ(B)=(B)=e020.(10.40)[0,T][0,T]dPtMoreover,B(t)=W(t)+q(s)ds,whereW(t)isaQ-Brownianmotion.0Proof:BytheRadon-Nikodymtheorem,(dQ/dP)=Λwith0<Λ<∞.SinceP(Λ>0)=1andEPΛ=1,existenceoftheprocessq(t)followsbyTheorem10.19.RepresentationforB(t)asaBrownianmotionwithdriftunderQfollowsfrom(10.27)inGirsanov’stheorem.
(Corollary10.21)isusedinFinance,whereq(t)denotesthemarketpriceforrisk.10.5ChangeofMeasureforPointProcessesConsiderapointprocessN(t)withintensityλ(t)(seeChapter9fordefini-tions).Thispresumesaprobabilityspace,whichcanbetakenasthespaceofright-continuousnon-decreasingfunctionswithunitjumps,andaproba-bilitymeasurePsothatNhasintensityλ(t)underP.Girsanov’stheoremassertsthatthereisaprobabilitymeasureQ,equivalenttoP,underwhichN(t)isPoissonprocesswiththeunitrate.Thusanequivalentchangeofmea-surecorrespondstoachangeintheintensity.Ifwelookuponthecompensatortλ(s)dsasthedrift,thenanequivalentchangeofmeasureresultsinachange0ofthedrift. 10.5.CHANGEOFMEASUREFORPOINTPROCESSES281Theorem10.22(Girsanov’stheoremforPoissonprocesses)LetN(t)beaPoissonprocesswithrate1underP,0≤t≤T.Foraconstantλ>0defineQbydQ(1−λ)T+N(T)lnλ=e.(10.41)dPThenunderQ,NisaPoissonprocesswithrateλ.Proof:LetΛ(t)=e(1−λ)t+N(t)lnλ=e(1−λ)tλN(t).(10.42)ItiseasytoseethatΛ(t)isaP-martingalewithrespecttothenaturalfiltrationoftheprocessN(t).Indeed,usingindependenceofincrementsofthePoissonprocess,wehaveE(Λ(t)|F)=e(1−λ)tE(λN(t)|F)PsPs=e(1−λ)tλN(s)E(λN(t)−N(s)|F)Ps=e(1−λ)tλN(s)E(λN(t)−N(s))=e(1−λ)tλN(s)e(λ−1)(t−s)=e(1−λ)sλN(s)=Λ(s).WehaveusedthatP-distributionofN(t)−N(s)isPoissonwithparameter(t−s),henceE(λN(t)−N(s))=e(λ−1)(t−s).WeshowthatunderQtheincre-PmentsN(t)−N(s)areindependentofthepastandhavethePoissonλ(t−s)distribution,establishingtheresult.Fixu>0.Theconditionalexpectationunderthenewmeasure,Theorem10.10,andthedefinitionofΛ(t)(10.42)yield)*u(N(t)−N(s))u(N(t)−N(s))Λ(t)EQe|Fs=EPe|FsΛ(s)=e(1−λ)(t−s)Ee(u+logλ)(N(t)−N(s))|FPs(1−λ)(t−s)(u+logλ)(N(t)−N(s))=eEPe)uλ(t−s)(e−1)=e.
Theorem10.23LetN(t),0≤t≤T,beaPoissonprocesswithrateλunderQ.DefinePbydP(λ−1)T−N(T)lnλ=e.(10.43)dQThenunderP,N(t)isaPoissonprocesswithrate1.AsacorollaryweobtainthatthemeasuresofPoissonprocesseswithcon-stantratesareequivalentwithLikelihoodratiogivenbythefollowingtheorem. 282CHAPTER10.CHANGEOFPROBABILITYMEASURETheorem10.24LetNbeaPoissonprocesswithrateλ>0undertheproba-bilityPλ.ThenitisaPoissonprocesswithrateµundertheequivalentmeasurePµdefinedbydPλ(µ−λ)T−N(T)(lnλ−lnµ)=e.(10.44)dPµNotethatthekeypointinchangingmeasurewasthemartingaleproperty(undertheoriginalmeasure)oftheLikelihoodΛ(t).Thispropertywases-tablisheddirectlyintheproofs.ObservethattheLikelihoodisthestochasticexponentialofthepointprocessmartingaleM(t)=N(t)−A(t).Forexample,inchangingtheratefrom1toλ,Λ=E(λ−1)MwithM(t)=N(t)−t.ItturnsoutthatageneralpointprocessN(t)withintensityλ(t)canbeobtainedfromaPoissonprocesswithrate1byachangeofmeasure.Infact,Theorem10.24holdsforgeneralpointprocesseswithstochasticintensities.TheformofstochasticexponentialE(λ−1)·Mfornon-constantλisnothardtoobtain,seeExercise10.8,TT(1−λ(s))ds+lnλ(s)dN(s)Λ=E(λ(s)−1)dM(s)(T)=e00(10.45)ThenextresultestablishesthatapointprocessisaPoissonprocess(withaconstantrate)underasuitablechangeofmeasure.Theorem10.25LetN(t)beaPoissonprocesswithrate1underP,andM(t)=N(t)−t.Ifforapredictableprocessλ(s),E(λ(s)−1)dM(s)isamartingalefor0≤t≤T,thenundertheprobabilitymeasureQdefinedby(10.45)dQ/dP=Λ,Nisapointprocesswiththestochasticintensityλ(t).Conversely,ifQisabsolutelycontinuouswithrespecttoPthenthereexistsapredictableprocessλ(t),suchthatunderQ,Nisapointprocesswiththestochasticintensityλ(t).10.6LikelihoodFunctionsWhenobservationsXaremadefromcompetingmodelsdescribedbytheprob-abilitiesPandQ,theLikelihoodistheRadon-NikodymderivativeΛ=dQ/dP.LikelihoodforDiscreteObservationsSupposethatweobserveadiscreterandomvariableX,andtherearetwocompetingmodelsforX:itcancomefromdistributionPorQ.FortheobservednumberxtheLikelihoodisgivenbyQ(X=x)Λ(x)=.(10.46)P(X=x) 10.6.LIKELIHOODFUNCTIONS283SmallvaluesofΛ(x)provideevidenceformodelP,andlargevaluesformodelQ.IfXisacontinuousrandomvariablewithdensitiesf0(x)underPandf1(x)underQ,thendP=f0(x)dx,dQ=f1(x)dx,andtheLikelihoodisgivenbyf1(x)Λ(x)=.(10.47)f0(x)Iftheobserveddataareafinitenumberofobservationsx1,x2,...,xnthensimilarlywithx=(x1,x2,...,xn)Q(X=x)f1(x)Λ(x)=,orΛ(x)=,(10.48)P(X=x)f0(x)dependingonwhetherthemodelsarediscreteorcontinuous.Notethatifonemodeliscontinuousandtheotherisdiscrete,thenthecorrespondingmeasuresaresingularandtheLikelihooddoesnotexist.LikelihoodRatiosforDiffusionsLetXbeadiffusionsolvingtheSDEwithaP-BrownianmotionB(t)dX(t)=µ1(X(t),t)dt+σ(X(t),t)dB(t),(10.49)SupposethatitsatisfiesanotherequationwithaQ-BrownianmotionW(t)dX(t)=µ2(X(t),t)dt+σ(X(t),t)dW(t).(10.50)TheLikelihood,aswehaveseen,isgivenby(10.33)dQTµ2(X(t),t)−µ1(X(t),t)1T(µ2(X(t),t)−µ1(X(t),t))2dB(t)−dtΛ(X)==e0σ(X(t),t)20σ2(X(t),t).[0,T]dP(10.51)SinceB(t)isnotobserveddirectly,theLikelihoodshouldbeexpressedasafunctionoftheobservedpathX[0,T].Usingequation(10.49)weobtaindX(t)−µ1(X(t),t)dtdB(t)=,andputtingthisintotheLikelihood,weobtainσ(X(t),t)22dQTµ2(X(t),t)−µ1(X(t),t)1Tµ2(X(t),t)−µ1(X(t),t)dX(t)−dtΛ(X)==e0σ2(X(t),t)20σ2(X(t),t).(10.52)TdPUsingtheLikelihoodadecisioncanbemadeastowhatmodelismoreappro-priateforX.Example10.5:(Hypothesestesting)Supposethatweobserveacontinuousfunctionxt,0≤t≤T,andwewanttoknowwhetheritisjustWhitenoise,thenullhypothesiscorrespondingtoprobability 284CHAPTER10.CHANGEOFPROBABILITYMEASUREmeasurePoritissomesignalcontaminatedbynoise,thealternativehypothesiscorrespondingtoprobabilitymeasureQ.H0:noise:dX(t)=dB(t),and,H1:signal+noise:dX(t)=h(t)dt+dB(t)Herewehaveµ1(x)=0,µ2(x)=h(t),σ(x)=1.TheLikelihoodisgivenbyTTdQh(t)dX(t)−1h2(t)dt2Λ(X)T==e00.(10.53)dPThisleadstotheLikelihoodratiotestoftheform:concludethepresenceofnoiseifΛ≥k,wherekisdeterminedfromsettingtheprobabilityofthetypeoneerrortoα.TTh(t)dB(t)−1h2(t)dt2P(e00≥k)=α.(10.54)Example10.6:(EstimationinOrnstein-UhlenbeckModel)ConsiderestimationofthefrictionparameterαintheOrnstein-Uhlenbeckmodelon0≤t≤T,dX(t)=−αX(t)dt+σdB(t).DenotebyPαthemeasurecorrespondingtoX(t),sothatP0correspondstoσB(t),0≤t≤T.TheLikelihoodisgivenby)*TT22dPα−αX(t)1αX(t)Λ(α,X[0,T])=dP=exp2dX(t)−2dt.0σ2σ00MaximizinglogLikelihood,wefindTX(t)dX(t)0αˆ=−.(10.55)TX2(t)dt0Remark10.3:LetX(t)andY(t)satisfythestochasticdifferentialequationsfor0≤t≤TdX(t)=µX(X(t),t)dt+σX(X(t),t)dW(t),(10.56)anddY(t)=µY(Y(t),t)dt+σY(Y(t),t)dW(t).(10.57)Considerprobabilitymeasuresinducedbythesediffusionsonthespaceofcontinuousfunctionson[0,T],C[0,T].ItturnsoutthatifσX=σYthenPXandPYaresingular(“live”ondifferentsets).Itmeansthatbyobservingaprocesscontinuouslyoveranintervaloftimewecandecidepreciselyfromwhichequationitcomes.Thisidentificationcanbemadewiththehelpofthequadraticvariationprocess,22d[X,X](t)d[X,X](t)=σ(X(t),t)dt,andσ(X(t),t)=,(10.58)dt 10.7.EXERCISES285and[X,X]isknownexactlyifapathisobservedcontinuously.IfσX=σY=σthenPXandPYareequivalent.TheRadon-Nikodymderivatives(theLikelihoods)aregivenbyTµ(X(t),t)−µ(X(t),t)Tµ2(X(t),t)−µ2(X(t),t)dPYYXdX(t)−1YXdt(X)=e0σ2(X(t),t)20σ2(X(t),t),[0,T]dPX(10.59)andTµ(Y(t),t)−µ(Y(t),t)Tµ2(Y(t),t)−µ2(Y(t),t)dPX−YXdY(t)+1YXdt(Y)=e0σ2(Y(t),t)20σ2(Y(t),t).[0,T]dPY(10.60)Notes.MaterialforthischapterisbasedonKaratzasandShreve(1988),LiptserandShirayev(1974),LambertonandLapeyre(1996).10.7ExercisesExercise10.1:LetPbeN(µ1,1)andQbeN(µ2,1)onIR.ShowthattheyareequivalentandthattheRadon-NikodymderivativedQ/dP=Λisgiven(µ1222−µ1)x+(µ−µ)byΛ(x)=e212.GivealsodP/dQ.Exercise10.2:ShowthatifXhasN(µ,1)distributionunderP,thenthereisanequivalentmeasureQ,suchthatXhasN(0,1)distributionunderQ.GivetheLikelihooddQ/dPandalsodP/dQ.GivetheQ-distributionofY=X−µ.Exercise10.3:YhasaLognormalLN(µ,σ2)distribution.UsingachangeofmeasurecalculateEYI(Y>K).Hint:changemeasurefromN(µ,σ2)toN(µ+σ2,σ2).Exercise10.4:LetX(t)=B(t)+sintforaP-BrownianmotionB(t).LetQbeanequivalentmeasuretoPsuchX(t)isaQ-Brownianmotion.GiveΛ=dQ/dP.Exercise10.5:LetBbeann-dimBrownianmotionandHanadaptednTiiregularprocess.LetH·B(T)=H(s)dB(s)beamartingale.Showi=10''1T'H(s)'2thatthemartingaleexponentialisgivenbyexp(H·B(T)−ds).20T2Hint:showthatquadraticvariationofH·Bisgivenby|H(s)|ds,where0|H(s)|2denotesthelengthofvectorH(s).Exercise10.6:LetWµ(t)=µt+B(t),whereB(t)isaP-Brownianmotion.Showthat−2y(y−x)P(maxWµ(t)≤y|Wµ(T)=x)=1−eT,x≤y,t≤T 286CHAPTER10.CHANGEOFPROBABILITYMEASUREand−2y(y−x)P(minWµ(t)≥y|Wµ(T)=x)=1−eT,x≥y.t≤THint:usethejointdistributions(10.38)and(3.16).Exercise10.7:ProveTheorem10.23.Exercise10.8:LetN(t)bePoissonprocesswithrate1andN¯(t)=N(t)−t.Showthatforanadapted,continuousandboundedprocessH(t),theprocesstM(t)=H(s)dN¯(s)isamartingalefor0≤t≤T.Showthat0tt−H(s)ds+ln(1+H(s))dN(s)E(M)(t)=e00.Exercise10.9:(Estimationofparameters)FindtheLikelihoodcorrespondingtodifferentvaluesofµoftheprocessX(t)givenbydX(t)=µX(t)dt+σX(t)dB(t)on[0,T].GivethemaximumLikeli-hoodestimator.Exercise10.10:VerifythemartingalepropertyoftheLikelihoodoccurringinthechangeofrateinaPoissonprocess.Exercise10.11:LetdQ=ΛdPonFT,Λ(t)=EP(Λ|Ft)iscontinuous.ForaP-martingaleM(t)findafinitevariationprocessA(t)suchthatM(t)=M(t)−A(t)isaQ-localmartingale.Exercise10.12:LetBandNberespectivelyaBrownianmotionandaPoissonprocessonthesamespace(Ω,F,IF,P),0≤t≤T.DefineΛ(t)=e(ln2)N(t)−tanddQ=Λ(T)dP.ShowthatBisaBrownianmotionunderQ.Exercise10.13:1.LetBandNberespectivelyaBrownianmotionandaPoissonprocessonthesamespace(Ω,F,IF,P),0≤t≤T,andX(t)=B(t)+N(t).GiveanequivalentprobabilitymeasureQ1suchthatB(t)+tandN(t)−tareQ1-martingales.DeducethatX(t)isaQ1-martingale.2.GiveanequivalentprobabilitymeasureQ2suchthatB(t)+2tandN(t)−2tareQ2-martingales.DeducethatX(t)isaQ2-martingale.3.DeducethatthereareinfinitelymanyequivalentprobabilitymeasuresQsuchthatX(t)=B(t)+N(t)isaQ-martingale. Chapter11ApplicationsinFinance:StockandFXOptionsInthischapterthefundamentalsofMathematicsofOptionPricingaregiven.Theconceptofarbitrageisintroduced,andamartingalecharacterizationofmodelsthatdon’tadmitarbitrageisgiven,theFirstFundamentalTheoremofassetpricing.Thetheoryofpricingbyno-arbitrageispresentedfirstintheFiniteMarketmodel,andtheninageneralSemimartingaleModel,wherethemartingalerepresentationpropertyisused.ChangeofmeasureanditsapplicationasthechangeofnumerairearegivenasacorollarytoGirsanov’stheoremandgeneralBayesformulaforexpectations.Theyrepresentthemaintechniquesusedforpricingforeignexchangeoptions,exoticoptions(asian,lookback,barrieroptions)andinterestratesoptions.11.1FinancialDerivativesandArbitrageAfinancialderivativeoracontingentclaimonanassetisacontractthatallowspurchaseorsaleofthisassetinthefutureontermsthatarespecifiedinthecontract.Anoptiononstockisabasicexample.Definition11.1AcalloptiononstockisacontractthatgivesitsholdertherighttobuythisstockinthefutureatthepriceKwritteninthecontract,calledtheexercisepriceorthestrike.AEuropeancalloptionallowstheholdertoexercisethecontract(thatis,tobuythisstockatK)ataparticulardateT,calledthematurityortheexpirationdate.AnAmericanoptionallowstheholdertoexercisethecontractatanytimebeforeoratT.287 288APPLICATIONSINFINANCEAcontractthatgivesitsholdertherighttosellstockiscalledaputoption.Therearetwopartiestoacontract,theseller(writer)andthebuyer(holder).Theholderoftheoptionshastherightbutnoobligationsinexercisingthecontract.Thewriteroftheoptionshastheobligationtoabidebythecontract,forexample,theymustsellthestockatpriceKifthecalloptionisexercised.Denotethepriceoftheasset(e.g.stock)attimetbyS(t).Acontingentclaimonthisassethasitsvalueatmaturityspecifiedinthecontract,anditissomefunctionofthevaluesS(t),0≤t≤T.SimplecontractsdependonlyonthevalueatmaturityS(T).Example11.1:(ValueatmaturityofEuropeancallandputoptions)Ifyouholdacalloption,thenyoucanbuyoneshareofstockatTforK.IfattimeT,S(T)H,K>H.0≤t≤T1T3.AsianOptions.PayoffatTdependsontheaveragepriceS¯=S(u)duT0+duringthelifeoftheoption.AveragecallpaysX=(S¯−K),andaverageput++X=(K−S¯).RandomstrikeoptionpaysX=(S(T)−S¯).Whereasthevalueofafinancialclaimatmaturitycanbeobtainedfromthetermsspecifiedinthecontract,itisnotsoclearhowtoobtainitsvaluepriortomaturity.Thisisdonebythepricingtheory,whichisalsousedtomanagefinancialrisk.Ifthepayoffofanoptiondependsonlyonthepriceofstockatexpiration,thenitispossibletographthevalueofanoptionatexpirationagainsttheunderlyingpriceofstock.Forexample,thepayofffunctionofacalloptionis(x−K)+.SomepayofffunctionsaregiveninExercise11.1. 11.1.FINANCIALDERIVATIVESANDARBITRAGE289ArbitrageandFairPriceArbitrageisdefinedinFinanceasastrategythatallowstomakeaprofitoutofnothingwithouttakinganyrisk.Mathematicalformulationofarbitrageisgivenlaterinthecontextofamarketmodel.Example11.3:(Futures,orForward)Consideracontractthatgivestheholder1shareofstockattimeT,soitsvalueattimeTisthemarketpriceS(T).Denotethepriceofthiscontractattime0byC(0).ThefollowingargumentshowsthatC(0)=S(0),asanyothervalueresultsinanarbitrageprofit.IfC(0)>S(0),thensellthecontractandreceiveC(0);buythestockandpayS(0).ThedifferenceC(0)−S(0)>0canbeinvestedinarisk-freebankaccountwithinterestrater.AttimeTyouhavethestocktodeliver,andmakearbitragerTprofitof(C(0)−S(0))e.IfC(0)0inarisk-freebankaccount.AttimeTexercisetherTcontractbybuyingbackstockatS(T).Theprofitis(S(0)−C(0))e.ThusanypriceC(0)differentfromS(0)resultsinarbitrageprofit.TheonlycasewhichdoesnotresultinarbitrageprofitisC(0)=S(0).ThefairpriceinagameofchancewithaprofitX(negativeprofitisloss)istheexpectedprofitEX.Theaboveexamplealsodemonstratesthatfinancialderivativesarenotpricedbytheirexpectations,andingeneral,theirarbitrage-freevalueisdifferenttotheirfairprice.Thefollowingexampleusesatwo-pointdistributionforthestockonexpi-rationtoillustratethatpricesforoptionscannotbetakenasexpectedpayoffs,andallthepricesbutone,leadtoarbitrageopportunities.Example11.4:Thecurrentpriceofthestockis$10.WewanttopriceacalloptiononthisstockwiththeexercisepriceK=10andexpirationinoneperiod.Supposethatthestockpriceattheendoftheperiodcanhaveonlytwovalues$8and$12pershare,andthattherisklessinterestrateis10%.Supposethecallispricedat$1pershare.Considerthestrategy:buycalloptionon200sharesandsell100sharesofstockST=12ST=8Buyoptionon200shares-2004000Sell(short)100shares1000-1200-800Savingsaccount800880880Profit0+80+80WecanseethatineithercaseST=8orST=12anarbitrageprofitof$80isrealized.Itcanbeseen,byreversingtheabovestrategy,thatanypriceabove$1.36willresultinarbitrage.Thepricethatdoesnotleadtoarbitrageis$1.36. 290APPLICATIONSINFINANCEEquivalencePortfolio.PricingbyNoArbitrageThemainideainpricingbyno-arbitrageargumentsistoreplicatethepayoffoftheoptionatmaturitybyaportfolioconsistingofstockandbond(cash).Toavoidarbitrage,thepriceoftheoptionatanyothertimemustequalthevalueofthereplicatingportfolio,whichisvaluedbythemarket.ConsidertheoneperiodcaseT=1first,andassumethatinoneperiodstockpricemovesupbyfactoruordownbyd,d<1j)C=SBin(j;n,p)−r−nKBin(j;n,p),nwherej=[ln(K/Sd)]+1andp=up.ln(u/d)rItispossibletoobtaintheoptionpricingformulaofBlackandScholesfromtheBinomialformulabytakinglimitsasthelengthofthetradingperiodgoestozeroandthenumberoftradingperiodsngoestoinfinity.PricingbyNoArbitrageGivenageneralmodelthesearethequestionweask.1.Ifwehaveamodelforevolutionofprices,howcanwetelliftherearearbitrageopportunities?Notfindinganyisagoodstart,butnotaproofthattherearenone.2.Ifweknowthattherearenoarbitrageopportunitiesinthemarket,howdowepriceaclaim,suchasanoption?3.Canwepriceanyoption,oraretheresomethatcannotbepricedbyarbitragearguments?Theanswersaregivenbytwomainresults,calledtheFundamentalTheoremsofassetpricing.Inwhatfollowsweoutlinethemathematicaltheoryofpricingofclaimsinfiniteandgeneralmarketmodels. 11.2.AFINITEMARKETMODEL29311.2AFiniteMarketModelConsideramodelwithonestockwithpriceS(t)attimet,andarisklessinvestment(bond,orcashinasavingsaccount)withpriceβ(t)attimet.Iftherisklessrateofinvestmentisaconstantr>1thenβ(t)=rtβ(0).AmarketmodeliscalledfiniteifS(t),t=0,...,Ttakefinitelymanyvalues.Aportfolio(a(t),b(t))isthenumberofsharesofstockandbondunitsheldduring[t−1,t).Theinformationavailableafterobservingpricesuptotimetisdenotedbytheσ-fieldFt.Theportfolioisdecidedonthebasisofinformationattimet−1,inotherwordsa(t),b(t)areFt−1measurable,t=1,...,T,orinourterminologytheyarepredictableprocesses.Thechangeinmarketvalueoftheportfolioattimetisthedifferencebetweenitsvalueafterithasbeenestablishedattimet−1anditsvalueafterpricesattareobserved,namelya(t)S(t)+b(t)β(t)−a(t)S(t−1)−b(t)β(t−1)=a(t)∆S(t)+b(t)∆β(t).Aportfolio(tradingstrategy)iscalledself-financingifallthechangesintheportfolioareduetogainsrealizedoninvestment,thatis,nofundsareborrowedorwithdrawnfromtheportfolioatanytime,
tV(t)=V(0)+a(i)∆S(i)+b(i)∆β(i)),t=1,2,...,T.(11.8)i=1Theinitialvalueoftheportfolio(a(t),b(t))isV(0)=a(1)S(0)+b(1)β(0)andsubsequentvaluesV(t),t=1,...,TaregivenbyV(t)=a(t)S(t)+b(t)β(t).(11.9)V(t)representsthevalueoftheportfoliojustbeforetimettransactionsaftertimetpricewasobserved.Sincethemarketvalueoftheportfolio(a(t),b(t))attimetafterS(t)isannouncedisa(t)S(t)+b(t)β(t),andthevalueofthenewlysetupportfolioisa(t+1)S(t)+b(t+1)β(t),aself-financingstrategymustsatisfya(t)S(t)+b(t)β(t)=a(t+1)S(t)+b(t+1)β(t).(11.10)Astrategyiscalledadmissibleifitisself-financingandthecorrespondingvalueprocessisnon-negative.Acontingentclaimisanon-negativerandomvariableXon(Ω,FT).ItrepresentsanagreementwhichpaysXattimeT,forexample,foracallwithstrikeK,X=(S(T)−K)+.Definition11.2AclaimXiscalledattainableifthereexistsanadmissiblestrategyreplicatingtheclaim,thatis,V(t)satisfies(11.8),V(t)≥0andV(T)=X. 294APPLICATIONSINFINANCEDefinition11.3AnarbitrageopportunityisanadmissibletradingstrategysuchthatV(0)=0,butEV(T)>0.NotethatsinceV(T)≥0,EV(T)>0isequivalenttoP(V(T)>0)>0.Thefollowingresultiscentraltothetheory.Theorem11.4SupposethereisaprobabilitymeasureQ,suchthatthedis-countedstockprocessZ(t)=S(t)/β(t)isaQ-martingale.Thenforanyad-missibletradingstrategythediscountedvalueprocessV(t)/β(t)isalsoaQ-martingale.SuchQiscalledanequivalentmartingalemeasure(EMM)orarisk-neutralprobabilitymeasure.Proof:Sincethemarketisfinite,thevalueprocessV(t)takesonlyfinitelymanyvalues,thereforeEV(t)exist.Themartingalepropertyisverifiedasfollows.)*V(t+1)'EQ'Ft=EQa(t+1)Z(t+1)+b(t+1)|Ftβ(t+1)=a(t+1)EQZ(t+1)|Ft+b(t+1)sincea(t)andb(t)arepredictable=a(t+1)Z(t)+b(t+1)sinceZ(t)isamartingale=a(t)Z(t)+b(t)since(a(t),b(t))isself-financing(11.10)V(t)=.β(t)
Thisresultis“nearly”theconditionforno-arbitrage,sinceitstatesthatifwestartwithzerowealth,thenpositivewealthcannotbecreatedifprob-abilitiesareassignedbyQ.Indeed,bytheaboveresultV(0)=0impliesEQ(V(T))=0,andQ(V(T)=0)=1.However,inthedefinitionofanarbi-tragestrategytheexpectationistakenundertheoriginalprobabilitymeasureP.ToestablishtheresultforPequivalenceofprobabilitymeasuresisused.RecallthedefinitionfromChapter10.Definition11.5TwoprobabilitymeasuresPandQarecalledequivalentiftheyhavesamenullsets,thatis,foranysetAwithP(A)=0,Q(A)=0andviceversa.Equivalentprobabilitymeasuresinamarketmodelreflectthefactthatin-vestorsagreeonthespaceofallpossibleoutcomes,butassigndifferentprob-abilitiestotheoutcomes.Thefollowingresultgivesaprobabilisticconditiontoassurethatthemodeldoesnotallowforarbitrageopportunities.ItstatesthatnoarbitrageisequivalenttoexistenceofEMM. 11.2.AFINITEMARKETMODEL295Theorem11.6(FirstFundamentalTheorem)AmarketmodeldoesnothavearbitrageopportunitiesifandonlyifthereexistsaprobabilitymeasureQ,equivalenttoP,suchthatthediscountedstockprocessZ(t)=S(t)/β(t)isaQ-martingale.Proof:Proofofsufficiency.SupposethereexistsaprobabilitymeasureQ,equivalenttoP,suchthatthediscountedstockprocessZ(t)=S(t)/β(t)isaQ-martingale.Thentherearenoarbitrageopportunities.ByTheorem11.4anyadmissiblestrategywithV(0)=0musthaveQ(V(T)>0)=0.SinceQisequivalenttoP,P(V(T)>0)=0.ButthenEP(V(T))=0.ThustherearenoadmissiblestrategieswithV(0)=0andEP(V(T))>0,inotherwords,thereisnoarbitrage.Aproofofnecessityrequiresadditionalconcepts,see,forexample,HarrisonandPliska(1981).
Claimsarepricedbythereplicatingportfolios.Theorem11.7(PricingbyNo-Arbitrage)Supposethatthemarketmodeldoesnotadmitarbitrage,andXisanattainableclaimwithmaturityT.ThenC(t),thearbitrage-freepriceofXattimet≤T,isgivenbyV(t),thevalueofaportfolioofanyadmissiblestrategyreplicatingX.Moreover)*β(t)'C(t)=V(t)=EQX'Ft,(11.11)β(T)whereQisanequivalentmartingaleprobabilitymeasure.Proof:SinceXisattainable,itisreplicatedbyanadmissiblestrategywiththevalueofthereplicatingportfolioV(t),0≤t≤T,andX=V(T).Fixonesuchstrategy.Toavoidarbitrage,thepriceofXatanytimetK)=ES(T)I(S(T)>K)−KQ(S(T)>K).ThesecondtermiseasytocalculatebyusingNormalprobabilitiesasK(1−Φ((logK−µ)/σ)).Thefirsttermcanbecalculatedbyadirectintegration,orbychangingmeasure,seeExample10.2.S(T)isabsorbedintotheLikelihooddQ1/dQ=Λ=S(T)/ES(T).Now,byLognormalityofS(T)writeitaseY,withY∼2N(µ,σ2).ThenES(T)=EeY=eµ+σ/2,andE(eYI(eY>K))=EeYE((eY/EeY)I(eY>K))=EeYE(ΛI(eY>K))=EeYEI(eY>K)=EeYQ(eY>K).Q112SincedQ/dQ=eY−µ−σ/2,itfollowsbyTheorem10.4thatQis11N(µ+σ2,σ2)distribution.Therefore2E(eYI(eY>K))=eµ+σ/2(1−Φ((lnK−µ−σ2)/σ)). 11.4.DIFFUSIONANDTHEBLACK-SCHOLESMODEL305Finally,using1−Φ(x)=Φ(−x),weobtain2+µ+σ2/2µ+σ−lnKµ−lnKE(X−K)=eΦ()−KΦ().(11.33)σσThepriceattimetoftheEuropeancalloptiononstockwithstrikeKandmaturityTisgivenby(Theorem11.13)−r(T−t)+C(t)=eEQ((S(T)−K)|Ft).Itfollowsfrom(11.33)thatC(t)isgivenbytheBlack-Scholesformula√−r(T−t)C(t)=S(t)Φ(h(t))−KeΦh(t)−σT−t,(11.34)wherelnS(t)+(r+1σ2)(T−t)h(t)=K√2.(11.35)σT−tPricingofClaimsbyaPDE.ReplicatingPortfolioLetXbeaclaimoftheformX=g(S(T)).SincethestockpricesatisfiesSDE(11.30),bytheMarkovpropertyofS(t)itfollowsfrom(11.32)thatthepriceofXattimetC(t)=e−r(T−t)Eg(S(T))|F=e−r(T−t)Eg(S(T))|S(t).(11.36)QtQBytheFeynman-Kacformula(Theorem6.8),C(x,t)=e−r(T−t)Eg(S(T))|S(t)=xQsolvesthefollowingpartialdifferentialequation(PDE)1∂2C(x,t)∂C(x,t)∂C(x,t)22σx+rx+−rC=0.(11.37)2∂x2∂x∂tTheboundaryconditionisgivenbythevalueoftheclaimatmaturityC(x,T)=g(x),x≥0,withg(x)isthevalueoftheclaimattimeTwhenthestockpriceisx.Whenx=0equation(11.37)gives−r(T−t)C(0,t)=eg(0),0≤t≤T,(portfolioofonlyabondhasitsvalueattimetasthediscountedfinalvalue.)Foracalloptiong(x)=(x−K)+.Thispartialdifferentialequationwassolved 306APPLICATIONSINFINANCEbyBlackandScholeswiththeFouriertransformmethod.ThesolutionistheBlack-Scholesformula(11.34).Nextwegiveanargumenttofindthereplicatingportfolio,whichisalsousedtore-derivethePDE(11.37).ItisclearthatC(t)in(11.36)isafunctionofS(t)andt.Let(a(t),b(t))beareplicatingportfolio.Sinceitisself-financingitsvalueprocesssatisfiesdV(t)=a(t)dS(t)+b(t)dβ(t).(11.38)IfXisanattainableclaim,thatis,X=V(T),thenthearbitrage-freevalueofXattimetK)−EKI(S(T)>K).Qβ(T)Qβ(T)Qβ(T)Thefirsttermisevaluatedbychangingthenumeraire,ES(T)I(S(T)>K)=E1S(T)I(S(T)>K)=β(0)Q(S(T)>K).Qβ(T)Q1ΛTβ(T)S(0)1ThesecondtermisKQ(S(T)>K)/β(T),whenβ(t)isdeterministic.Thusβ(0)KC=Q1(S(T)>K)−Q(S(T)>K).(11.51)S(0)β(T)ThisisageneralizationoftheBlack-Scholesformula(Gemanetal.(1995),Bj¨ork(1998),Klebaner(2002)).OnecanverifythattheBlack-Scholesformulaisalsoobtainedwhenthestockisusedasthenumeraire(Exercise11.12).Ifβ(t)isstochastic,thenbyausingT-bondasanumeraire,)*KEQI(S(T)>K)=P(0,T)KQT(S(T)>K),β(T)whereP(0,T)=E(1/β(T)),Λ(T)=1.ToevaluateexpectationsQP(0,T)β(T)above,weneedtofindthedistributionunderthenewmeasure.ThiscanbedonebyusingtheSDEunderQ1.SDEsunderaChangeofNumeraireLetS(t)andβ(t)bepositiveprocesses.LetQandQ1berespectivelytheequivalentmartingalemeasureswhenβ(t)andS(t)arenumeraires,sothatS(t)/β(t)isaQ-martingale,andβ(t)/S(t)aQ1-martingale.SupposethatunderQdS(t)=µS(t)dt+σS(t)dB(t),dβ(t)=µβ(t)dt+σβ(t)dB(t),whereBisaQ-Brownianmotion,andcoefficientsareadaptedprocesses.LetX(t)havetheSDEsunderQandQ,withaQ-BrownianmotionB111dX(t)=µ0(X(t))dt+σ(X(t))dB(t),dX(t)=µ(X(t))dt+σ(X(t))dB1(t),1Sincethesemeasuresareequivalent,thediffusioncoefficientforXisthesame,andthedriftcoefficientsarerelatedbyTheorem11.18)*σS(t)σβ(t)µ1(X(t))=µ0(X(t))+σ(X(t))−.(11.52)S(t)β(t) 312APPLICATIONSINFINANCEProof:WeknowfromSection10.3,thatQ1andQarerelatedby)*·dQ1Λ(T)==EH(t)dB(t)(T),dQ0µ1(X(t))−µ0(X(t))dQ1withH(t)=.Ontheotherhand,byTheorem11.17=σ(X(t))dQS(T)/S(0)S(t)/S(0)Λ(T)=β(T)/β(0).MoreoverΛ(t)=EQ(Λ(T)|Ft)=β(t)/β(0)bythemartin-galepropertyof(S/β).UsingtheexponentialSDE,)*S(t)/S(0)dΛ(t)=Λ(t)H(t)dB(t)=d,β(t)/β(0)itfollowsthat)*µ1(X(t))−µ0(X(t))β(t)S(t)dB(t)=d.σ(X(t))S(t)β(t)UsingtheSDEsforSandβ,andthatS/βhasnodtterm,theresultfollows.
Corollary11.19TheSDEforS(t)underQ1,whenitisanumeraire,is)*σ2(t)dS(t)=µ(t)+Sdt+σ(t)dB1(t),(11.53)SSS(t)whereB1(t)isaQ-martingale.Moreover,intheBlack-Scholesmodelwhen1µ=µS,andσ=σS,thenewdriftcoefficientis(µ+σ2)S.SSForotherresultsonchangeofnumeraireseeGemanetal.(1995).11.6Currency(FX)OptionsCurrencyoptionsinvolveatleasttwomarkets,onedomesticandforeignmar-kets.Forsimplicityweconsiderjustoneforeignmarket.Fordetailsonarbi-tragetheoryinForeignMarketDerivativesseeMusielaandRutkowski(1998).Theforeignanddomesticinterestratesinrisklessaccountsaredenotedbyrfandrd,(subscriptsfandddenotingforeignanddomesticrespectively).TheEMM’sexistinbothmarkets,QfandQd.LetU(t)denotethevalueofaforeignassetinforeigncurrencyattimet,sayJPY.AssumethatU(t)evolvesaccordingtheBlack-Scholesmodel,andwriteitsequationundertheEMM(risk-neutral)measure,dU(t)=rfU(t)dt+σUU(t)dBf(t),(11.54) 11.6.CURRENCY(FX)OPTIONS313whereBfisaQf-Brownianmotion.AsimilarequationholdsforassetsinthedomesticmarketunderitsEMMQ.ThepriceprocessdiscountedbyerdtdinthedomesticmarketisaQd-martingale,andanoptiononanassetinthedomesticmarketwhichpaysC(T)(indomesticcurrency)atThasitstimetpriceC(t)=e−rd(T−t)E(C(T)|F).QdtAlinkbetweenthetwomarketsisprovidedbytheexchangerate.TakeX(t)tobethepriceinthedomesticcurrencyof1unitofforeigncurrencyattimet.NotethatX(t)itselfisnotanasset,butitgivesthepriceofaforeignassetindomesticcurrencywhenmultipliedbythepriceoftheforeignasset.WeassumethatX(t)followstheBlack-ScholesSDEunderthedomesticEMMQddX(t)=µXX(t)dt+σXX(t)dBd(t),andshowthatµmustber−r.Taketheforeignsavingsaccounterft,itsXdfvalueindomesticcurrencyisX(t)erft.Thuse−rdtX(t)erftshouldbeaQ-dmartingale,asadiscountedpriceprocessinthedomesticmarket.Butfrom(11.6)−rtrt(r−r+µ12edX(t)ef=S(0)efdX−2σX)t+σXBd(t),implyingµX=rd−rf.ThustheequationforX(t)underthedomesticEMMQdisdX(t)=(rd−rf)X(t)dt+σXX(t)dBd(t),(11.55)withaQd-BrownianmotionBd(t).Ingeneral,theBrownianmotionsBdandBfarecorrelated,i.e.forsome|ρ|<1,Bf(t)=ρBd(t)+1−ρ2W(t),withindependentBrownianmotionsBdandW.d[Bd,W](t)=ρdt.(Themodelcanalsobesetupbyusingtwo-dimensionalindependentBrownianmotions,a(column)vectorQd-Brownianmotion[Bd,W],andamatrixσU,therowvectorσU=[ρ,1−ρ2]).ThevalueoftheforeignassetindomesticcurrencyattimetisgivenbyU(t)X(t)=U˜(t).Thereforee−rdtU˜(t),asadiscountedpriceprocess,isaQd-martingale.Onecaneasilyseefrom(11.54)and(11.55)that−rt−1(σ2+σ2)t+σedU˜(t)=X(0)U(0)e2UXUBf(t)+σXBd(t).ForittobeaQd-martingale,itmustholddBf(t)+ρσXdt=dB˜d(t)isaQd-Brownianmotion(correlatedwithBd).ThisisaccomplishedbyGirsanov’stheorem,andtheSDEforU(t)underthedomesticEMMQdbecomesdU(t)=(rf−ρσUσX)U(t)dt+σUU(t)dB˜d(t).(11.56)NowitisalsoeasytoseethattheprocessU˜(t)(theforeignassetindomesticcurrency)underthedomesticEMMQdhastheSDEdU˜(t)=rdU˜(t)dt+(σUdB˜d(t)+σXdBd(t))U˜(t).(11.57) 314APPLICATIONSINFINANCESinceσUdB˜d(t)+σXdBd(t)=¯σdW˜d(t),forsomeQd-BrownianmotionW˜d(t),withσ¯2=σ2+σ2+2ρσσ,(11.58)UXUXthevolatilityofU˜(t)isgivenbytheformulathatcombinestheforeignassetvolatilityandtheexchangeratevolatility(11.58).OptionsonForeignCurrencyConsideracalloptionononeunitofforeigncurrency,sayonJPYwiththestrikepriceKinAUD.ItspayoffatTinAUDis(X(T)−K)+.ThusitsddpriceattimetisgivenbyC(t)=e−rd(T−t)E((X(T)−K)+|F).QddtTheconditionalQd-distributionofX(T)givenX(t)isLognormal,byus-ing(11.6),andstandardcalculationsgivetheBlack-Scholescurrencyformula(Garman-Kohlhagen(1983))√C(t)=X(t)e−rf(T−t)Φ(h(t))−Ke−rd(T−t)Φ(h(t)−σT−t),dXlnX(t)+(r−r+σ2/2)(T−t)KddfSh(t)=√.σXT−tOtheroptionsoncurrencyarepricedsimilarly,byusingequation(11.6).OptionsonForeignAssetsStruckinForeignCurrencyConsideroptionsonaforeignassetdenominatedinthedomesticcurrency.AcalloptionpaysattimeTtheamountX(T)(U(T)−K)+.SincethefamountisspecifiedinthedomesticcurrencythepricingshouldbedoneunderthedomesticEMMQd.Thefollowingargumentachievestheresult.IntheforeignmarketoptionsarepricedbytheBlack-Scholesformula.ThepriceindomesticcurrencyisobtainedbymultiplyingbyX(t).Acalloptionispricedby√C(t)=U(t)Φ(h(t))−Ke−rf(T−t)Φh(t)−σT−t,fUwherelnU(t)+(r+1σ2)(T−t)Kff2Uh(t)=√.σUT−tCalculationsofthepriceunderQdareleftasanexerciseinchangeofnu-meraire,Exercise11.13. 11.7.ASIAN,LOOKBACKANDBARRIEROPTIONS315OptionsonForeignAssetsStruckinDomesticCurrencyConsideracalloptionontheforeignassetU(t)withstrikepriceKdinAUD.ItspayoffatTinAUDis(X(T)U(T)−K)+=(U˜(T)−K)+.ThusitspriceddattimetisgivenbyC(t)=e−rd(T−t)E((U˜(T)−K)+|F).QddtTheconditionaldistributionofU˜(T)givenU˜(t)isLognormal,byequation(11.57),withvolatilitygivenby¯σ2=σ2+σ2+2ρσσfrom(11.58).ThusUXUX√C(t)=U˜(t)Φ(h(t))−Ke−rd(T−t)Φ(h(t)−σ¯T−t),dU˜(t)σ¯2lnKd+(rd+2)(T−t)h(t)=√.σ¯T−tGuaranteedExchangedRate(Quanto)OptionsAquantooptionpaysindomesticcurrencytheforeignpayoffconvertedatapredeterminedrate.ForexampleaquantocallpaysattimeTtheamountX(0)(U(T)−K)+,(Kisinforeigncurrency).ThusitstimetvalueisgivenffbyC(t)=X(0)e−rd(T−t)E((U(T)−K)+|F).QdftTheconditionaldistributionofU(T)givenU(t)isLognormal,byequation(11.56),withvolatilityσU.Standardcalculationsgive√C(t)=X(0)e−rd(T−t)U(t)eδ(T−t)Φ(h(t))−KΦ(h(t)−σT−t),fUσ2lnU(t)+(δ+U)(T−t)Kf2whereδ=rf−ρσUσXandh(t)=√.σUT−t11.7Asian,LookbackandBarrierOptionsAsianOptionsWeassumethatS(t)satisfiestheBlack-ScholesSDE,whichwewriteundertheEMMQdS(t)=rS(t)dt+σS(t)dB(t).+1TAsianoptionspayattimeT,theamountC(T)givenbyS(u)du−KT0+1T(fixedstrikeK)orS(t)dt−KS(T)(floatingstrikeKS(T)).BothT0TkindsinvolvetheintegralaverageofthestockpriceS¯=1S(u)du.TheT0 316APPLICATIONSINFINANCEaverageofLognormalsisnotananalyticallytractabledistribution,thereforedirectcalculationsdonotleadtoaclosedformsolutionforthepriceoftheoption.PricingbyusingPDEswasdonebyRogersandShi(1995),andVecer(2002).WepresentVecer’sapproach,whichrestsontheideathatitispossibletoreplicatetheintegralaverageofthestockS¯byaself-financingportfolio.Let1−r(T−t)a(t)=(1−e)andrTdV(t)=a(t)dS(t)+r(V(t)−a(t)S(t))dt=rV(t)dt+a(t)(dS(t)−rS(t)dt),(11.59)withV(0)=a(0)S(0)=1(1−e−rT)S(0).ItiseasytoseethatV(t)isarTself-financingportfolio(Exercise11.14).SolvingtheSDE(11.59)(bylookingatd(V(t)e−rt))forV(t)andusingintegrationbyparts,weobtainTrTrTr(T−t)V(T)=eV(0)+a(T)S(T)−ea(0)S(0)−eS(t)da(t)0T1=S(t)dt,(11.60)T0becauseofd(er(T−t)S(t)a(t))=er(T−t)a(t)dS(t)−rer(T−t)S(t)dt+er(T−t)S(t)da(t),andTTa(T)S(T)−erTa(0)S(0)=er(T−t)a(t)(dS(t)−rS(t)dt)+er(T−t)S(t)da(t).00e−rTtTheself-financingportfolioV(t)consistsofa(t)sharesandb(t)=S(u)duT0cashandhasthevalueS¯attimeT.Considernextpricingtheoptionwiththepayoff(S¯−KS(T)−K)+,12whichencompassesboththefixedandthefloatingkinds(bytakingoneofK’sas0).Toreplicatesuchanoptionholdattimeta(t)ofthestock,startwithinitialwealthV(0)=a(0)S(0)−e−rTKandfollowtheself-financingstrategy2(11.59).TheterminalvalueofthisportfolioisV(T)=S¯−K2.Thepayoffoftheoptionis(S¯−KS(T)−K)+=(V(T)−KS(T))+.121Thusthepriceoftheoptionattimetis−r(T−t)+C(t)=eEQ(V(T)−K1S(T))|Ft).LetZ(t)=V(t)/S(t).Thenproceeding,C(t)=e−r(T−t)E(S(T)(Z(T)−K)+|F).Q1t 11.7.ASIAN,LOOKBACKANDBARRIEROPTIONS317ThemultiplierS(T)insideisdealtwithbythechangeofnumeraire,andtheremainingexpectationisthatofadiffusionZ(t),whichsatisfiesthebackwardPDE.UsingS(t)asanumeraire(Theorem11.17,(11.49))wehaveC(t)=S(t)E((Z−K)+|F).(11.61)Q1TtByIto’sformulatheSDEforZ(t)underQ,whenertisanumeraire,is2dZ(t)=σ(Z(t)−a(t))dt+σ(a(t)−Z(t))dB(t).ByTheorem11.18theSDEforZ(t)undertheEMMQ1,whenS(t)isanumeraire,isdZ(t)=−σ(Z(t)−q(t))dB1(t),whereB1(t)isaQ-Brownianmotion,(B1(t)=B(t)+σt).Nowwecanwrite1aPDEfortheconditionalexpectationin(11.61).E((Z(T)−K)+|F)=E((Z(T)−K)+|Z(t)),Q1tQ11bytheMarkovproperty.Henceu(x,t)=E((Z(T)−K)+|Z(t)=x)Q11satisfiesthePDE(seeTheorem6.6)σ2∂2u∂u2(x−a(t))+=0,2∂x2∂tsubjecttotheboundaryconditionu(x,T)=(x−K)+.Finally,thepriceof1theoptionattimezeroisgivenbyusing(11.61)V(0,S(0),K1,K2)=S(0)u(0,Z(0)),withZ(0)=V(0)/S(0)=1(1−e−rT)−e−rTK/S(0).rT2LookbackOptionsAlookbackcallpaysX=S(T)−SandalookbackputX=S∗−S(T),∗whereSandS∗denotethesmallestandthelargestvaluesofstockon[0,T].∗Thepriceofalookbackputisgivenby−rT∗−rT∗−rTC=eEQ(S−S(T))=eEQ(S)−eEQ(S(T)),(11.62)whereQisthemartingaleprobabilitymeasure.SinceS(t)e−rtisaQ-martingale,e−rTE(S(T))=S(0).TofindE(S∗),QQtheQ-distributionofS∗isneeded.TheequationforS(t)isgivenbyS(t)= 318APPLICATIONSINFINANCE(r−1σ2)t+σB(t)∗S(0)e2withaQ-BrownianmotionB(t),see(11.31).Clearly,Ssatisfies∗(r−1σ2)t+σB(t)∗S=S(0)e2.ThedistributionofmaximumofaBrownianmotionwithdriftisfoundbyachangeofmeasure,seeExercise10.6andExample10.4(10.38).BytheGirsanov’stheorem(r−1σ2)t+σB(t)=σW(t),foraQ-Brownianmotion21dQcW(T)−1c2T12∗W(t),with(W[0,T])=e2,c=(r−σ)/σ.Clearly,S=dQ12∗S(0)eW.ThedistributionofW∗underQisthedistributionofthemaximum1ofBrownianmotion,anditsdistributionunderQisobtainedasinExample10.4(see(10.38)).Therefore∞−rT∗−rTσyeEQ(S)=eS(0)efW∗(y)dy,(11.63)−∞wherefW∗(y)isobtainedfrom(10.38)(seealsoExercise11.15).Lookbackcallsarepricedsimilarly.ThepriceofaLookbackcallisgivenbyS(0)times22σ2ln(K/S(0))+σTσ2ln(K/S(0))−σTσ2(1+)Φ√2−(1−)e−rTΦ√2−.2rσT2rσT2r(11.64)BarrierOptionsExamplesofBarrierOptionsaredown-and-outcallsordown-and-incalls.Therearealsocorrespondingputoptions.Adown-and-outcallgetsknockedoutwhenthepricefallsbelowacertainlevelH,ithasthepayoff(S(T)−K)+I(S≥H),S(0)>H,K>H.Adown-and-incallhasthepayoff∗(S(T)−K)+I(S≤H).TopricetheseoptionsthejointdistributionofS(T)∗andS∗isneededunderQ.Forexample,thepriceofdown-and-incallisgivenby−rT+C=eEQ((S(T)−K)I(S∗≤H))∞ln(H/S(0))σ=e−rTS(0)eσx−Kg(x,y)dxdy,ln(K/S(0))−∞σwhereg(x,y)istheprobabilitydensityof(W(T),W∗(T))underthemartingalemeasureQ.ItisfoundbychangingmeasureasdescribedaboveandinExample10.4(seealsoExercise11.16).DoublebarrieroptionshavepayoffsdependingonS(T),SandS∗.Anexampleisacallwhichpaysonlyifthepricenever∗goesbelowacertainlevelH1oraboveacertainlevelH2withthepayoffX=(S(T)−K)+I(H0impliesEQ(X)>0,andviceversa.Exercise11.5:(Pricinginincompletemarkets)1.ShowthatifM(t),0≤t≤T,isamartingaleundertwodifferentprob-abilitymeasuresQandP,thenforsttheprocessstartsatr(t)andrunsfors−t,givings−tr(s)=a∗−e−b(s−t)(a∗−r(t))+σe−b(s−t)ebudB(u),(12.29)0 330APPLICATIONSINFINANCEwithaQ-BrownianmotionB(u),independentofFt.Thusr(s)isaGaussianprocess,sincethefunctionintheItˆointegralisdeterministic(seeTheoremT4.11).Theconditionaldistributionoftr(s)dsgivenFtisthesameasthatgivenr(t)andisaNormaldistribution.ThusthecalculationofthebondpriceT−r(s)dsinvolvestheexpectationofaLognormal,EQ(et|r(t)).Thus2P(t,T)=e−µ1+σ1/2,(12.30)Twhereµandσ2aretheconditionalmeanandvarianceofr(s)dsgiven11tr(t).Using(12.29)andcalculatingdirectlyorusing(12.4)h=a∗−h/b,wehave'TT'µ1=EQr(s)ds'r(t)=EQ(r(s)|r(t))dstt∗1∗−b(T−t)=a(T−t)+(a−r(t))(1−e).(12.31)bTocalculateσ2,usetherepresentationforr(s)conditionalonr(t),equation1(12.29)TTTTσ2=Covr(s)ds,r(u)du=Cov(r(s),r(u))dsdu.1ttttNowitisnothardtocalculate,bytheformulaforthecovarianceofItˆoGaus-sianintegrals(4.26)or(4.27),thatfors>u2σb(u−s)−b(u+s−2t)Cov(r(s),r(u))=e−e.2bPuttingthisexpression(andasimilaroneforwhens0,theQ-conditionaldistributionofP(T,T+δ)givenFtisLognormalwithmeanTP(t,T+δ)122EQ(lnP(T,T+δ)|Ft)=ln−(τ(s,T+δ)−τ(s,T))ds,P(t,T)2tandvarianceTγ2(t)=Var(lnP(T,T+δ)|F)=(τ(s,T+δ)−τ(s,T))2ds.(12.57)ttProof:Bylettingt=T1=TandT2=T+δinequation(12.49),theexpressionforP(T,T+δ)isobtained,P(0,T+δ)T1T22−(τ(s,T+δ)−τ(s,T))dB(s)−(τ(s,T+δ)−τ(s,T))dsP(T,T+δ)=e020.P(0,T)(12.58)TofinditsconditionaldistributiongivenFt,separatetheFt-measurableterm,P(0,T+δ)t1t22−(τ(s,T+δ)−τ(s,T))dB(s)−(τ(s,T+δ)−τ(s,T))dsP(T,T+δ)=e020P(0,T)T1T22−(τ(s,T+δ)−τ(s,T))dB(s)−(τ(s,T+δ)−τ(s,T))ds×et2tP(t,T+δ)T1T22−(τ(s,T+δ)−τ(s,T))dB(s)−(τ(s,T+δ)−τ(s,T))ds=et2t.(12.59)P(t,T)Ifτ(t,T)isnon-random,thentheexponentialtermisindependentofFt.HencethedesiredconditionaldistributiongivenFtisLognormalwiththemeanandvarianceasstated.
Corollary12.8TheconditionaldistributionofP(T,T+δ)givenFtundertheforwardmeasureQT+δisLognormalwithmeanTP(t,T+δ)12EQT+δ(lnP(T,T+δ)|Ft)=ln+(τ(s,T+δ)−τ(s,T))ds,P(t,T)2tandvarianceγ2(t)(12.57).Proof:Useequation(12.59)fortheconditionalrepresentation,togetherwith(12.54),aQ-BrownianmotiondBT+δ(t)=dB(t)+τ(t,T+δ)dt,toT+δhaveP(t,T+δ)TT+δ1T2−(τ(s,T+δ)−τ(s,T))dB(s)+(τ(s,T+δ)−τ(s,T))dsP(T,T+δ)=et2t.P(t,T)(12.60)
Bytakingt=T=0,T+δ=TinTheorem12.7anditscorollaryweobtain 12.7.OPTIONS,CAPSANDFLOORS339Corollary12.9InthecaseofconstantvolatilitiestheQ-distributionandQT-distributionofP(t,T)areLognormal.TheLognormalityofP(t,T)canalsobeobtaineddirectlyfromequations(12.47)and(12.38).12.7Options,CapsandFloorsWegivecommonoptionsoninterestratesandshowhowtheyrelatetooptionsonbonds.CapandCapletsAcapisacontractthatgivesitsholdertherighttopaytherateofinterestsmallerofthetwo,thefloatingrate,andratek,specifiedinthecontract.Apartyholdingthecapwillneverpayrateexceedingk,therateofpaymentiscappedatk.SincethepaymentsaredoneatasequenceofpaymentsdatesT,T,...,T,calledatenor,withT=T+δ(e.g.δ=1ofayear),the12ni+1i4rateiscappedoverintervalsoftimeoflengthδ.Thusacapisacollectionofcaplets.fitTTTTTT012ii+1nFigure12.3:Paymentdatesandsimplerates.Consideracapletover[T,T+δ].Withoutthecaplet,theholderofaloanmustpayattimeT+δaninterestpaymentoffδ,wherefisthefloating,simplerateovertheinterval[T,T+δ].Iff>k,thenacapletallowstheholdertopaykδ.Thusthecapletisworthfδ−kδattimeT+δ.Iffk,and(12.81)ik.Usingtheaboverelationiterativelyfromitok,weobtain
i−1TiTkL(t,Tj)δdB(t)=dB(t)+γ(t,Tj)dt.(12.83)1+L(t,Tj)δj=kReplaceBTkintheSDE(12.79)forL(t,T)underQ,k−1TkdL(t,T)=γ(t,T)L(t,T)dBTk(t),byBTiwiththedriftfrom(12.83)k−1k−1k−1toobtain(12.80).ThecaseiK|Ft)−KP(t,s)Qs(P(s,T)>K|Ft),wheresistheexercisetimeoftheoptionandQs,QTaresandT-forwardmeasures.Exercise12.2:ShowthataEuropeancalloptiononthebondintheMertonmodelisgivenby2222P(t,T)σ(T−s)(s−t)P(t,T)σ(T−s)(s−t)lnKP(t,s)+2lnKP(t,s)−2P(t,T)Φ√−KP(t,s)Φ√.σ(T−s)s−tσ(T−s)s−tExercise12.3:(StochasticFubiniTheorem)LetH(t,s)becontinuous0≤t,s≤T,andforanyfixeds,H(t,s)asaprocessint,0≤t≤TisadaptedtotheBrownianmotionfiltrationFt.AssumeT2TH(t,s)dt<∞,sothatforeachstheItˆointegralX(s)=H(t,s)dW(t)00Tisdefined.SinceH(t,s)iscontinuous,Y(t)=H(t,s)dsisdefinedanditis0continuousandadapted.AssumeTT2EH(t,s)dtds<∞.00TTT1/21.ShowthatE|X(s)|ds≤EH2(t,s)dtds<∞,consequently000TX(s)dsexists.0 348APPLICATIONSINFINANCE2.If0=t00,andthatv2(t)=1d[k,k](s)istk2(s)deterministic.ShowthatSwaption(t)=b(t)(α+(t)k(t)−kα−(t)),where)*lnk(t)/kv(t)α±(t)=Φ±.v(t)2 Thispageintentionallyleftblank Chapter13ApplicationsinBiologyInthischapterapplicationsofstochasticcalculustopopulationmodelsaregiven.Diffusionmodels,MarkovJumpprocessmodels,age-dependentnon-Markovmodelsandstochasticmodelsforcompetitionofspeciesarepresented.DiffusionModelsareusedinvariousareasofBiologyasmodelsforpopulationgrowthandgeneticevolution.BirthandDeathprocessesarerandomwalksincontinuoustimeandaremetinvariousapplications.Anovelapproachtotheage-dependentbranching(Bellman-Harris)processisgivenbytreatingitasasimplemeasure-valuedprocess.ThestochasticmodelforinteractingpopulationsthatgeneralizestheLotka-Volterraprey-predatormodelistreatedbyusingasemimartingalerepresentation.Itispossibletoformulatethesemodelsasstochasticdifferentialorintegralequations.Wedemonstratehowresultsonstochasticdifferentialequationsandmartingalespresentedinearlierchaptersareappliedfortheiranalysis.13.1Feller’sBranchingDiffusionAsimplebranchingprocessisamodelinwhichindividualsreproduceinde-pendentlyofeachotherandofthehistoryoftheprocess.Thecontinuousapproximationtobranchingprocessisthebranchingdiffusion.ItisgivenbythestochasticdifferentialequationforthepopulationsizeX(t),00.ThenP(X(t)≥0forallt≥0)=1,moreoverifτ=inf{t:X(t)=0},thenX(t)=0forallt≥τ.Proof:Considerfirst(13.1)withX(0)=0.Clearly,X(t)≡0isasolution.ConditionsoftheYamada-Watanaberesultontheexistenceanduniquenessaresatisfied,seeTheorem5.5.Thereforesolutionisunique,andX(t)≡0istheonlysolution.Considernow(13.1)withX(0)>0.ThefirsthittingtimeofzeroτisastoppingtimewithX(τ)=0.BythestrongMarkovpropertyX(t)fort>τisthesolutionto(13.1)withzeroinitialcondition,henceitiszeroforallt>τ.ThusifX(0)>0,theprocessispositiveforallt<τ,andzeroforallt≥τ.
Thenextresultdescribestheexponentialgrowthofthepopulation.Theorem13.2LetX(t)solve(13.1)andX(0)>0.ThenEX(t)=X(0)eαt.X(t)e−αtisanon-negativemartingalewhichconvergesalmostsurelytoanon-degeneratelimitast→∞.Proof:FirstweshowthatX(t)isintegrable.SinceItˆointegralsarelocaltmartingales,X(s)dB(s)isalocalmartingale.LetTnbealocalizing0t∧Tnsequence,sothatX(s)dB(s)isamartingaleintforanyfixedn.0Thenusing(13.1)wecanwritet∧Tnt∧TnX(t∧Tn)=X(0)+αX(s)ds+σX(s)dB(s).(13.3)00Takingexpectation,weobtaint∧TnEX(t∧Tn)=X(0)+αEX(s)ds.(13.4)0t∧TntSinceXisnon-negative,X(s)dsisincreasingtoX(s)ds.Therefore00bymonotoneconvergence,t∧TntEX(s)ds→EX(s)ds,00 13.1.FELLER’SBRANCHINGDIFFUSION353tasn→∞.Thereforelimn→∞EX(t∧Tn)=X(0)+E0X(s)ds.SinceX(t∧Tn)→X(t)weobtainbyFatou’slemma(notingthatifalimitexists,thenliminf=lim)EX(t)=ElimX(t∧Tn)=EliminfX(t∧Tn)n→∞n→∞t≤liminfEX(t∧Tn)=limEX(t∧Tn)=X(0)+EX(s)ds.n→∞n→∞0UsingGronwall’sinequality(Theorem1.20)weobtainthattheexpectationisfinite,EX(t)≤(1+X(0))eαt.(13.5)tNowwecanshowthatthelocalmartingaleX(s)dB(s)isatruemartin-0gale.Consideritsquadraticvariation>··?tEX(s)dB(s),X(s)dB(s)(t)=EX(s)ds0.Thentheprobability−2αxofultimateextinctioniseσ2ifα>0,and1ifα≤0Proof:LetT0=τ=inf{t:X(t)=0}bethefirsttimeofhittingzero,andTbthefirsttimeofhittingb>0.Bytheformula(6.52)onexitprobabilitiesS(b)−S(x)Px(T00,Px(T00.AnalysisofthismodelisdonebyusingthediffusionprocessestechniquesdescribedinChapter6,butitistooinvolvedtobegivenhereindetail.TheimportantfeatureofthismodelisthatfixationdoesnotoccurandX(t)admitsastationarydistribution.Stationarydistributionscanbefoundbyformula(6.69).WefindxC2µ(y)2γ1−12γ2−1π(x)=expdy=C(1−x)x,(13.16)σ2(x)xσ2(y)0whichisthedensityofaBetadistribution.C=Γ(2γ1)Γ(2γ2)/Γ(2γ1+2γ2).Diffusionmodelsfindfrequentuseinpopulationgenetics,seeforexam-ple,Ewens(1979).FormoreinformationonWright-Fisherdiffusionsee,forexample,KarlinandTaylor(1981),Chapter15.Remark13.1:TheTheoryofweakconvergenceisusedtoshowthatadiffu-sionapproximationtoallelesfrequenciesisvalid.Thistheoryisnotpresentedherebutcanbefoundinmanytexts,seeforexampleLiptserandShiryayev(1989),EthierandKurtz(1986). 356CHAPTER13.APPLICATIONSINBIOLOGY13.3Birth-DeathProcessesABirth-Deathprocessisamodelofadevelopingintimepopulationofpar-ticles.Eachparticlelivesforarandomlengthoftimeattheendofwhichitsplitsintotwoparticlesordies.Iftherearexparticlesinthepopulation,thenthelifespanofaparticlehasanexponentialdistributionwithparametera(x),thesplitintotwooccurswithprobabilityp(x)andwiththecomplimentaryprobability1−p(x)aparticlediesproducingnooffspring.DenotebyX(t)thepopulationsizeattimet.Thechangeinthepopulationoccursonlyatthedeathofaparticle,andfromstatextheprocessjumpstox+1ifasplitoccursortox−1ifaparticledieswithoutsplitting.Thusthejumpvariablesξ(x)takevalues1and−1withprobabilitiesp(x)and1−p(x)respectively.Usingthefactthattheminimumofexponentiallydistributedrandomvariablesisexponentiallydistributed,wecanseethattheprocessstaysatxforanexponentiallydistributedlengthoftimewithparameterλ(x)=xa(x).Usually,theBirth-Deathprocessisdescribedintermsofbirthanddeathrates;inapopulationofsizex,aparticleisbornatrateb(x)anddiesattherated(x).Theserefertotheinfinitesimalprobabilitiesofpopulationincreasinganddecreasingbyone,namelyforanintegerxP(X(t+δ)=x+1|X(t)=x)=b(x)δ+o(δ),P(X(t+δ)=x−1|X(t)=x)=d(x)δ+o(δ),andwiththecomplimentaryprobabilitynobirthsordeathshappenin(t,t+δ)andtheprocessremainsunchangedP(X(t+δ)=x|X(t)=x)=1−(b(x)+d(x))δ+o(δ).Hereo(δ)denotesafunctionthatlimδ→0o(δ)/δ=0.Oncepopulationreacheszeroitstaysthereforever.P(X(t+δ)=0|X(t)=0)=1.ItcanbeseenthattheseassumptionsleadtothemodelofaJumpMarkovprocesswiththeholdingtimeparameteratxλ(x)=b(x)+d(x),(13.17)andthejumpfromx@b(x)+1withprobabilityb(x)+d(x)ξ(x)=d(x)(13.18)−1withprobabilityb(x)+d(x) 13.3.BIRTH-DEATHPROCESSES357Thefirsttwomomentsofthejumpsareeasilycomputedtobeb(x)−d(x)m(x)=andv(x)=1.(13.19)b(x)+d(x)ByTheorem9.15thecompensatorofX(t)isgivenintermsofλ(x)m(x)=b(x)−d(x),whichisthesurvivalrateinthepopulation.ThustheBirth-DeathprocessX(t)canbewrittenastX(t)=X(0)+b(X(s))−d(X(s))ds+M(t),(13.20)0whereM(t)isalocalmartingalewiththesharpbrackettM,M(t)=b(X(s))+d(X(s))ds.(13.21)0SinceE|ξ(x)|=1,itfollowsbyTheorem9.16thatifthelineargrowthcondi-tionb(x)+d(x)≤C(1+x),(13.22)holdsthenthelocalmartingaleMinrepresentation(13.20)isasquareinte-grablemartingale.Sometimesitisconvenienttodescribethemodelintermsoftheindividualbirthanddeathrates.Inapopulationofsizex,eachparticleisbornattherateβ(x)anddiesattherateγ(x).Theindividualratesrelatetothepopulationratesbyb(x)=xβ(x),andd(x)=xγ(x).(13.23)Introducetheindividualsurvivalrateα(x)=β(x)−γ(x).Thenthestochasticequation(13.20)becomestX(t)=X(0)+α(X(s))X(s)ds+M(t).(13.24)0Birth-DeathProcesseswithLinearRatesSupposethattheindividualbirthanddeathratesβandγareconstants.Thiscorrespondstolinearratesb(x)=βx,d(x)=γx(see(13.23)).Thelineargrowthcondition13.22issatisfied,andwehavefromequation(13.20)tX(t)=X(0)+αX(s)ds+M(t),(13.25)0 358CHAPTER13.APPLICATIONSINBIOLOGYwhereα=β−γisthesurvivalrateandM(t)isamartingale.Moreover,by(13.21)tM,M(t)=(β+γ)X(s)ds.(13.26)0Takingexpectationsin(13.25)tEX(t)=X(0)+αEX(s)ds,(13.27)0andsolvingthisequation,wefindEX(t)=Xeαt.(13.28)0Usingintegrationbyparts(notethateαthaszerocovariationwithX(t)),wehavedX(t)e−αt=e−αtdX(t)−αe−αtX(t−)dt.(13.29)SinceX(t−)dtcanbereplacedbyX(t)dt,byusingtheequation(13.25)initsdifferentialformweobtaindX(t)e−αt=e−αtdX(t)−αe−αtX(t)dt=e−αtdM(t).(13.30)Thust−αt−αsX(t)e=X(0)+edM(s).(13.31)0Usingtheruleforthesharpbracketoftheintegralandequation(13.26),wefind2··3te−αsdM(s),e−αsdM(s)(t)=e−2αsdM,M(s)000t−2αs=(β+γ)eX(s)ds.0SinceEX(t)=X(0)eαt,bytakingexpectationsitfollowsthat2··3Ee−αsdM(s),e−αsdM(s)(∞)<∞.00ConsequentlyX(t)e−αtisasquareintegrablemartingaleon[0,∞).23tβ+γVar(X(t)e−αt)=Ee−αsdM(s)(t)=X(0)1−e−αt.(13.32)0αSinceX(t)e−αtissquareintegrableon[0,∞),itconvergesalmostsurelyast→∞toanon-degeneratelimit. 13.3.BIRTH-DEATHPROCESSES359ProcesseswithStabilizingReproductionConsiderthecasewhentheratesstabilizeaspopulationsizeincreases,namelyβ(x)→βandγ(x)→γasx→∞.Then,clearly,α(x)=β(x)−γ(x)→α=β−γ.Dependingonthevalueofαradicallydistinctmodesofbehaviouroccur.Ifα<0thentheprocessdiesoutwithprobabilityone.Ifα>0thentheprocesstendstoinfinitywithapositiveprobability.Iftherateofconvergenceofα(x)toαisfastenough,thenexponentialgrowthpersistsasintheclassicalcase.Letε(x)=α(x)−α.Thenundersometechnicalconditions,thefollowingconditionisnecessaryandsufficientforexponentialgrowth,∞ε(x)dx<∞.(13.33)1xIf(13.33)holdsthenX(t)e−αtconvergesalmostsurelyandinthemeansquaretoanon-degeneratelimit.Thecaseα=0providesexamplesofslowlygrowingpopulations,suchasthosewithalinearrateofgrowth.Letα(x)>0andα(x)↓0.Amongsuchprocessestherearesomethatbecomeextinctwithprobabilityone,butthereareothersthathaveapositiveprobabilityofgrowingtoinfinity.Considerthecasewhenα(x)=c/xwhenx>0.Itispossibletoshowthatifc>1/2thenq=P(X(t)→∞)>0.Stochasticequation(13.24)becomestX(t)=X(0)+cI(X(s)>0)ds+M(t).(13.34)0Bytakingexpectationsin(13.34)onecanseethatlimt→∞EX(t)/t=cq,andthemeanofsuchprocessesgrowslinearly.ByusingTheorem9.19andExercise9.5itispossibletoshowthatforanyk,E(Xk(t))/tkconvergestothek-thmomentofagammadistributiontimesq.Thisimpliesthatontheset{X(t)→∞},X(t)/tconvergesindistributiontoagammadistribution.Inotherwords,suchprocessesgrowlinearly,andnotexponentially.Bychangingtherateofconvergenceofα(x)otherpolynomialgrowthratesofthepopulationcanbeobtained.Similarresultsareavailableforpopulation-dependentMarkovBranchingprocessesthatgeneralizeBirth-Deathprocessesbyallowingarandomnumberofoffspringattheendofaparticle’slife.FordetailsofthestochasticequationapproachtoMarkovJumpprocessesseeKlebaner(1994),wheresuchprocesseswerestudiedasrandomlyperturbedlineardifferentialequations;HamzaandKlebaner(1995). 360CHAPTER13.APPLICATIONSINBIOLOGY13.4BranchingProcessesInthisSectionwelookatapopulationmodelknownastheage-dependentBellman-HarrisBranchingprocess,seeHarris(1963),AthreyaandNey(1972).ThismodelgeneralizestheBirthandDeathprocessmodelintworespects:firstlythelifespanofindividualsneednothavetheexponentialdistribution,andsecondly,morethanoneparticlecanbeborn.Intheage-dependentmodelaparticlelivesforarandomlengthoftimewithdistributionG,andattheendofitslifeleavesarandomnumberofchildrenY,withmeanm=EY.Allparticlesreproduceanddieindependentlyofeachother.DenotebyZ(t)thenumberofparticlesaliveattimet.Itcanbeseenthatunlessthelifetimedistributionisexponential,theprocessZ(t)isnotMarkovian,anditsanalysisisusuallydonebyusingRenewalTheory(Harris(1963)).Hereweapplystochasticcalculustoobtainalimitingresultforthepopulationsizecountedinaspecialway(byreproductivevalue).Consideracollectionofindividualswithages(a1,...,az)=A.Itisconve-nienttolookatthevectorofagesA=(a1,...,az)asacountingmeasureA,zi+definedbyA(B)=i=11B(a),foranyBorelsetBinIR.ForafunctionfonIRthefollowingnotationsareused
z(f,A)=f(x)A(dx)=f(ai).i=1ThepopulationsizeprocessinthisnotationisZ(t)=(1,At).InthisapproachtheprocessofagesAtisameasure-valuedprocess,althoughsimplerthanstudiede.g.inDawson(1993).Toconvertameasure-valuedprocessintoascalarvalued,testfunctionsareused.TestfunctionsusedonthespaceofmeasuresareoftheformF((f,µ)),whereFandfarefunctionsonIR.LetG(a)h(a)=berateofdyingatagea.Ewithandwithoutthesubscript1−G(a)Adenotestheexpectationwhentheprocessesstartswithindividualsaged(a1,...,az)=A.Thefollowingresultisobtainedbydirectcalculations.+Theorem13.4ForaboundeddifferentiablefunctionFonIRandacon-+tinuouslydifferentiablefunctionfonIR,thefollowinglimitexists%&1limEAF((f,At))−F((f,A))=GF((f,A)),(13.35)t→0tGF((f,A))=F((f,A))(f,A)(13.36)
z+h(aj){EF(Yf(0)+(f,A)−f(aj))−F((f,A))},Aji=1 13.4.BRANCHINGPROCESSES361TheoperatorGin(13.36)definesageneratorofameasure-valuedbranchingprocessinwhichthemovementoftheparticlesisdeterministic,namelyshift.Thefollowingresultgivestheintegralequation(SDE)fortheageprocessunderthetestfunctions.Theorem13.5ForaboundedC1functionFonIRandaC1functionfon+IRtF,fF((f,At))=F((f,A0))+GF((f,As))ds+Mt,(13.37)0F,fwhereMtisalocalmartingalewiththesharpbracketgivenby45ttMF,f,MF,f=GF2((f,A))ds−2F((f,A))GF((f,A))ds.(13.38)tsss00Consequently,ttE(MF,f)2=EGF2((f,A))ds−2F((f,A))GF((f,A))ds,AtAsss00providedE(MF,f)2exists.AtProof:ThefirststatementisobtainedbyDynkin’sformula.Expression(13.38)isobtainedbylettingUt=F((f,At))andanapplicationofthefol-lowingresult.
Theorem13.6LetUtbealocallysquare-integrablesemi-martingalesuchthatUt=U0+At+Mt,whereAtisapredictableprocessoflocallyfinitevariationandMisalocallysquare-integrablelocalmartingale,A=M=0.LetU2=t00tU2+B+N,whereBisapredictableprocessandNisalocalmartingale.0ttttThent
M,M=B−2UdA−(A−A)2.(13.39)tts−sss−0s≤tOfcourse,ifAiscontinuous(asinourapplications)thelasttermintheaboveformula(13.39)vanishes.Proof:Bythedefinitionofthequadraticvariationprocess(orintegrationbyparts)tttU2=U2+2UdU+[U,U]=U2+2UdA+2UdM+[U,U].t0s−st0s−ss−st000 362CHAPTER13.APPLICATIONSINBIOLOGYUsingtherepresentationforU2givenintheconditionsofthelemma,wettobtainthat[U,U]t−Bt+20Us−dAsisalocalmartingale.Since[A,A]t=(A−A)2,theresultfollows.s≤tss−
Letv2=E(Y2)bethesecondmomentoftheoffspringdistribution.Ap-plyingDynkin’sformulatothefunctionF(u)=u(andwritingMfforM1,f),weobtainthefollowingTheorem13.7ForaC1functionfonIR+tf(f,At)=(f,A0)+(Lf,As)ds+Mt,(13.40)0wherethelinearoperatorLisdefinedbyLf=f−hf+mhf(0),(13.41)fandMtisalocalsquareintegrablemartingalewiththesharpbracketgivenbyt45Mf,Mf=f2(0)v2h+hf2−2f(0)mhf,Ads.(13.42)ts0Proof:ThefirststatementisDynkin’sformulaforF(u)=u.Thisfunctionisunboundedandthestandardformulacannotbeapplieddirectly.Howeveritcanbeappliedbytakingsmoothboundedfunctionsthatagreewithuonboundedintervals,Fn(u)=uforu≤n,andthesequenceofstoppingtimesTn=inf{(f,At)>n}asalocalizingsequence.TheformoftheoperatorLfollowsfrom(13.36).Similarly(13.42)followsfrom(13.38)bytakingF(u)=u2.
Bytakingftobeaconstant,f(u)=1,Theorem13.7yieldsthefollowingcorollaryforthepopulationsizeattimet,Z(t)=(1,At).tCorollary13.8ThecompensatorofZ(t)isgivenby(m−1)0(h,As)ds.ItisusefultobeabletotakeexpectationsinDynkin’sformula.Thenextresultgivesasufficientcondition.Theorem13.9Letf≥0beaC1functiononIR+thatsatisfies|(Lf,A)|≤C(1+(f,A)),(13.43)forsomeconstantCandanyA,andassumethat(f,A0)isintegrable.Thenff(f,At)andMtin(13.37)arealsointegrablewithEMt=0. 13.4.BRANCHINGPROCESSES363Proof:LetTnbealocalizingsequence,thenfrom(13.37)t∧Tnf(f,At∧Tn)=(f,A0)+(Lf,As)ds+Mt∧Tn,(13.44)0fwhereMisamartingale.Takingexpectationswehavet∧Tnt∧TnE(f,At∧Tn)=E(f,A0)+E(Lf,As)ds.(13.45)0ByCondition(13.43),t∧Tnt∧Tn|E(Lf,As)ds|≤E|(Lf,As)|ds00t∧Tn≤Ct+CE(f,As)ds.0Thuswehavefrom(13.45)t∧TnE(f,At∧Tn)≤E(f,A0)+Ct+CE(f,As)ds0t≤E(f,A0)+Ct+CEI(s≤Tn)(f,As)ds0t≤E(f,A0)+Ct+CE(f,As∧Tn)ds.0ItnowfollowsbyGronwall’sinequality(Theorem1.20)thatE(f,A)≤E(f,A)+Ct+teCt<∞.(13.46)t∧Tn0Takingn→∞,weconcludethatE(f,At)<∞byFatou’slemma.Thus(f,At)isintegrable,asitisnon-negative.NowbyCondition(13.43)tttE|(Lf,As)ds|≤E|(Lf,As)|ds≤C(1+E(f,As))ds<∞.(13.47)000ttItfollowsfrom(13.47)that0(Lf,As)dsanditsvariationprocess0|(Lf,As)|dsarebothintegrable,andfrom(13.37)thattfMt=(f,At)−(f,A0)−(Lf,As)ds(13.48)0isintegrablewithzeromean.
+ForsimplicityweassumethatG(u)<1forallu∈IR.Equation(13.40)canbeanalyzedthroughtheeigenvalueproblemfortheoperatorL. 364CHAPTER13.APPLICATIONSINBIOLOGYTheorem13.10LetLbetheoperatorin(13.41).ThentheequationLq=rq(13.49)hasasolutionqrforanyr.Thecorrespondingeigenfunction(normedsothatq(0)=1)isgivenbyruume−rsqr(u)=1−edG(s).(13.50)1−G(u)0Proof:Sinceeigenfunctionsaredetermineduptoamultiplicativeconstant,wecantakeq(0)=1.Equation(13.49)isafirstorderlineardifferentialequation,andsolvingitweobtainthesolution(13.50).Theorem13.11LetqrbeapostiveeigenfunctionofLcorrespondingtotheeigenvaluer.ThenQ(t)=e−rt(q,A)isapositivemartingale.rrtProof:Using(13.40)andthefactthatqrisaneigenfunctionforL,wehavetqr(qr,At)=(qr,A0)+r(qr,As)ds+Mt,(13.51)0qrwhereMtisalocalmartingale.Thefunctionsqrclearlysatisfycondition(13.43).Therefore(qr,At)isintegrable,anditfollowsfrom(13.51)bytakingexpectationsthatE(q,A)=ertE(q,A).(13.52)rtr0Usingintegrationbypartsfore−rt(q,A),weobtainfrom(13.51)thatrtdQ(t)=d(e−rt(q,A))=e−rtdMqr,rrttandtQ(t)=(q,A)+e−rsdMqr(13.53)rr0s0isalocalmartingaleasanintegralwithrespecttothelocalmartingaleMqr.Sinceapositivelocalmartingaleisasupermartingale,andQr(t)≥0,Qr(t)isasuper-martingale.Butfrom(13.52)itfollowsthatQr(t)hasaconstantmean.ThusthesupermartingaleQr(t)isamartingale.
TheMalthusianparameterαisdefinedasthevalueofrwhichsatisfies∞me−rudG(u)=1.(13.54)0WeassumethattheMalthusianparameterαexistsandispositive,inthiscasetheprocessiscalledsupercritical. 13.4.BRANCHINGPROCESSES365Theorem13.12ThereisonlyoneboundedpositiveeigenfunctionV,thereproductivevaluefunction,correspondingtotheeigenvalueαwhichistheMalthusianparameter,αu∞me−αsV(u)=edG(s).(13.55)1−G(u)u∞Proof:Itfollowsthatforr>α,me−rudG(u)<1andtheeigenfunction0qrin(13.50)ispositiveandgrowsexponentiallyfastorfaster.Forr<α,theeigenfunctionqtakesnegativevalues.Whenr=α,qα=Vin(13.55).Torseethatitisboundedbym,replacee−αsbyitslargestvaluee−αu.
ThisisthemainresultTheorem13.13W=e−αt(V,A)isapositivesquareintegrablemartingale,ttandthereforeconvergesalmostsurelyandinL2tothenon-degeneratelimitW≥0,EW=(V,A0)>0andP(W>0)>0.Proof:ThatWtisamartingalefollowsbyTheorem13.11.ItispositivethereforeitconvergesalmostsurelytoalimitW≥0bythemartingalecon-vergenceTheorem7.11.Toshowthatthelimitisnotidenticallyzero,weshowthatconvergenceisalsoinL2,i.e.thesecondmoments(andthefirstmoments)converge,implyingthatEW=limt→∞EWt=EW0=(V,A0).Itfollowsfrom(13.42)that45tVV22M,M=(σ+(m−V))h,Asds.(13.56)t0By(13.53)tW=(V,A)+e−αsdMV,(13.57)t0s0andweobtainthatt45tW,W=e−2αsdMV,MV=e−2αs(σ2+(m−V)2)h,Ads.tss00(13.58)NowitiseasytocheckthatthereisaconstantCandr,α≤r<2α,suchthat(σ2+(m−V)2)h,A≤C(q,A),(13.59)srsandusingTheorem13.11,∞∞Ee−2αs(σ2+(m−V)2)h,Ads0:Xt∨Yt=∞}.ThePoissonprocesseswithstate-dependentratesin(13.63)areobtainedasfollowst
t
π=I(X≥n)dΠa(n),π=I(XY≥n)dΠb/K(n),ts−sts−s−s00n≥1n≥1t
t
c/KdπAt=I(Xs−Ys−≥n)dΠs(n),πAt=I(Ys−≥n)dΠs(n).00n≥1n≥1 13.5.STOCHASTICLOTKA-VOLTERRAMODEL369Itiseasytoseethatπ,π,πA,πAhavetherequiredproperties.Theirjumpsttttareofsizeone.SinceallPoissonprocessesareindependent,theirjumpsaredisjoint,sothatthejumpsofπ,π,πA,πAaredisjointaswell.Todescribetheirttttintensitiesintroduceastochasticbasis(Ω,F,IF=(Ft)t≥0,P),thefiltrationIFisgeneratedbyallPoissonprocessesandsatisfiesthegeneralconditions.Then,obviously,therandomprocesst
A=I(X≥n)adsts0n≥1ttisadaptedandcontinuous,hencepredictable.Usingf(s−)ds=f(s)ds,00t
t
aπt−At=I(Xs−≥n)dΠs(n)−I(Xs−≥n)ads00n≥1n≥1t
=I(Xs−≥n)dΠsa(n)−ads.0n≥1Thusπ−Aisanintegralwithrespecttoamartingale,henceisalocalttmartingale.ThereforeAisthecompensatorofπ.SinceXisaninteger-ttstvaluedrandomvariable,I(Xs≥n)=Xs,givingthatAt=0aXsdsandn≥1theintensityofπisaX,asclaimed.Analogously,othercompensatorsaretttttbcAt=XsYsds,AAt=XsYsds,AAt=dYsds,0K0K0andthusallotherintensitieshavetherequiredform.Wenowshowthattheprocess(Xt,Yt)t≥0doesnotexplode.Theorem13.14P(T∞=∞)=1.Proof:SetTX=inf{t>0:X≥n},n≥1anddenotebyTX=limTX.nt∞nn→∞Using(13.64)weobtaintEXt∧TX≤X0+aEXs∧TXds.nn0aTByGronwall’sinequality,EXTX∧T≤X0eforeveryT>0.HencebythenaTFatou’slemmaEXTX∧T≤X0e.Consequently,forallT>0∞P(TX≤T)=0.∞ 370CHAPTER13.APPLICATIONSINBIOLOGYSetTY=inf{t:Y≥},≥1anddenotebyTY=limTY.Usingt∞→∞(13.64)weobtaintcEYt∧TX∧TY≤Y0+EXs∧TX∧TYYs∧TX∧TYdsnKnn0tc≤Y0+nEYs∧TX∧TYds.Kn0cnTHence,byGronwall’sinequality,foreveryT>0,EYT∧Tx∧Ty≤Y0eandnmbytheFatou’slemma,cnTEYT∧TX∧TY≤Y0eK,n≥1.n∞Consequently,P(TY≤TX∧T)=0,∀T>0,n≥1and,sinceTX∞,as∞nnn→∞,weobtainP(TY≤T)=0.∞SinceT=TX∧TY,P(T≤T)≤P(TX≤T)+P(TY≤T),andwehave∞∞∞∞∞∞P(T∞≤T)=0foranyT>0.
Corollary13.15ForTn=inf{t:Xt∨Yt≥n},andforallT>0limP(Tn≤T)=0.n→∞Theabovedescriptionofthemodelallowsustoclaimthat(Xt,Yt)isacontinuous-timepurejumpMarkovprocesswithjumpsoftwopossiblesizesinbothcoordinates:“1”and“−1”andinfinitesimaltransitionprobabilities(asδt→0)'PXt+δt=Xt+1'Xt,Yt=aXtδt+o(δt)'bPXt+δt=Xt−1'Xt,Yt=XtYtδt+o(δt)K'cPYt+δt=Yt+1'Xt,Yt=XtYtδt+o(δt)K'PYt+δt=Yt−1'Xt,Yt=dYtδt+o(δt).Semimartingaledecompositionfor(xK,yK)ttLetA,A,AA,AAbethecompensatorsofπ,π,πA,πAdefinedabove.Intro-ttttttttducemartingalesM=π−A,M=π−A,MB=πA−AA,MB=πA−AA,tttttttttttt 13.5.STOCHASTICLOTKA-VOLTERRAMODEL371andalsonormalizedmartingalesM−MMB−MCmK=ttandmAK=tt.(13.65)ttKKThen,from(13.64)itfollowsthattheprocess(xK,yK)admitsthesemimartin-ttgaledecompositiontxK=x+[axK−bxKyK]ds+mK,t0ssst0tKKKKKyt=y0+[cxsys−dys]ds+mAt,(13.66)0whichisastochasticanalogue(inintegralform)oftheequations(13.61).Inthesequelweneedquadraticvariationsofthemartingalesin(13.66).ByTheorem9.3allmartingalesarelocallysquareintegrableandpossessthepredictablequadraticvariationsM,M=A,M,M=AandMB,MB=AA,MB,MB=AA,tttttttt(13.67)andzerocovariationsM,M≡0,...,MB,MB≡0,sincethejumpsofttπ,π,πA,πAaredisjoint.Henceweobtainthesharpbracketsofthemartin-ttttgalesinequations(13.66),mK,mAK≡0,tt1mK,mK=axK+bxKyKds,tsssK0t1mAK,mAK=cxKyK+dyKds.(13.68)tsssK0NotethatthestochasticequationsabovedonotsatisfythelineargrowthconditioninxK,yK.Nevertheless,thesolutionexistsforallt.ttDeterministic(Fluid)approximationWenowshowthattheLotka-Volterraequations(13.61)describealimit(alsoknownasfluidapproximation)forthefamily(xK,yK)asparameterK→∞.ttResultsonthefluidapproximationfordiscontinuousMarkovprocessescanbefoundinKurtz(1981).Theorem13.16ForanyT>0andη>0limPsup|xK−x|+|yK−y|>η=0.ttttK→∞t≤T 372CHAPTER13.APPLICATIONSINBIOLOGYProof:SetKKKTn=inf{t:xt∨yt≥n}.(13.69)ByCorollary13.15,sinceTK=T,nnKKlimlimsupPTn≤T=0.(13.70)n→∞K→∞Hence,itsufficestoshowthatforeveryn≥1,KKlimPsup|xt−xt|+|yt−yt|>η=0.(13.71)K→∞t≤TK∧TnSincesup(xK∨yK)≤n+1,thereisaconstantL,dependingonnandttnt≤TnK∧TT,suchthatfort≤TK∧Tn'''''''axK−bxKyK−ax−bxy'≤L'xK−x'+'yK−y'ttttttntttt'''''''cxKyK−dyK−cxy−dy'≤L'xK−x'+'yK−y'.ttttttnttttTheseinequalitiesand(13.61),(13.66)implyK−xK|xTK∧TTK∧T|+|yTK∧T−yTK∧T|nnnnt|xK−xK≤2Lns∧TKs∧TK|+|ys∧TK−ys∧TK|dsnnnn0KK+sup|mt|+sup|mAt|.t≤TnK∧Tt≤TnK∧TNow,byGronwall’sinequalitywefindKK2LnTKKsup|xt−xt|+|yt−yt|≤esup|mt|+sup|mAt|.t≤TK∧Tt≤TK∧Tt≤TK∧TnnnTherefore(13.71)holdsifbothsup|mK|andsup|mAK|convergeinttt≤TK∧Tt≤TK∧TnnprobabilitytozeroasK→∞.By(13.68)anddefinition(13.69)ofTK,nK12EmTK∧T≤(a(n+1)+b(n+1))T.nKThusbyDoob’sinequalityformartingales(8.47)|mK|)2K,mK→0.E(supt≤4EmTK∧Tnt≤TK∧Tn 13.6.EXERCISES373asK→∞.Thisimpliessup|mK|→0asK→∞.Thesecondtermt≤TK∧Ttnsup|mAK|istreatedsimilarly,andtheproofiscomplete.t≤TK∧Ttn
UsingthestochasticLotka-Volterramodelonecanevaluateasymptotically,whenKislarge,thetimetoextinctionofpreyspecies,aswellasthelikelytrajectorytoextinction.OnesuchtrajectoryisintheFigurebelow.Thisanal-ysisisdonebyusingtheLargeDeviationsTheory,seeKlebanerandLiptser(2001).10864200.20.40.60.8Figure13.2:Alikelypathtoextinction.13.6ExercisesExercise13.1:FindtheexpectedtimetoextinctionET0intheBranchingdiffusionwithα<0.Hint:useTheorem6.16andformula(6.98)tofindE(T0∧Tb).ET0=limb→∞E(T0∧Tb)bymonotoneconvergence.Exercise13.2:LetX(t)beabranchingdiffusionsatisfyingSDE(13.1).1.Letc=2α/σ2.Showthate−cXtisamartingale.2.LetTbethetimetoextinction,T=inf{t:Xt=0},whereT=∞ifXt>0forallt≥0,andletqt(x)=Px(T≤t)betheprobabilitythatextinctionoccursbytimetwhentheinitialpopulationsizeisx.Provethatq(x)≤e−cx.tExercise13.3:AmodelforpopulationgrowthisgivenbythefollowingSDE√dX(t)=2X(t)dt+X(t)dB(t),andX(0)=x>0.Findtheprobabilitythatthepopulationdoublesitsinitialsizexbeforeitbecomesextinct.Showthat 374CHAPTER13.APPLICATIONSINBIOLOGYwhentheinitialpopulationsizexissmallthenthisprobabilityisapproxi-mately1/2,butwhenxislargethisprobabilityisnearlyone.Exercise13.4:Checkthatthedistributionin(13.16)intheWright-Fisherdiffusionisstationary.Exercise13.5:Letadeterministicgrowthmodelbegivenbythedifferentialequationdx(t)=g(x(t))dt,x0>0,t≥0,forapositivefunctiong(x)andconsideritsstochasticanaloguedX(t)=g(X(t))dt+σ(X(t))dB(t),X(0)>0.Onewaytoanalyzethestochasticequationisbycomparisonwiththedeter-ministicsolution.1.FindG(x)suchthatG(x(t))=G(x(0))+tandconsiderY(t)=G(X(t)).FinddY(t).2.Letg(x)=xr,0≤r<1,andσ(x)/xr→0asx→∞.Giveconditionsong(x)andσ2(x)sothattheLawofLargeNumbersholdsforY(t),thatis,Y(t)/t→1,ast→∞ontheset{Y(t)→∞}.AsystematicanalysisofthismodelisgiveninKelleretal.(1987).Exercise13.6:LetL1,L2,...,Lxbeindependentexponentiallydistributedrandomvariableswithparametera(x),(theyrepresentthelifespanofparticlesinthepopulationofsizex).Showthatmin(L1,L2,...,Lx)hasanexponentialdistributionwithparameterxa(x).(Inabranchingmodelthechangeinthepopulationsizeoccurswhenaparticledies,thatis,atmin(L1,L2,...,Lx)).Exercise13.7:(Birth-Deathprocessesstochasticrepresentation)LetNλ(t)andNµ(t),k≥1,betwoindependentsequencesofindependentkkPoissonprocesseswithratesλandµ.LetX(0)>0andfort>0X(t)satisfiest
t
λµX(t)=X(0)+I(X(s−)≥k)dNk(s)−I(X(s−)≥k)dNk(s).00k≥1k≥11.ShowthatX(t)isaBirth-Deathprocessandidentifytherates.2.GivethesemimartingaledecompositionforX(t). Chapter14ApplicationsinEngineeringandPhysicsInthischaptermethodsofStochasticCalculusareappliedtotheFilteringprobleminEngineeringandRandomOscillatorsinPhysics.TheFilteringproblemconsistsoffindingthebestestimatorofasignalwhenobservationsarecontaminatedbynoise.ForanumberofclassicalequationsofmotionsinPhysicswefindstationarydensitieswhenthemotionissubjectedtorandomexcitations.14.1FilteringThefilteringproblemistheproblemofestimationofasignalcontaminatedbynoise.LetY(t)betheobservationprocess,andFYdenotetheinformationtavailablebyobservingtheprocessuptotimet,thatisFY=σ(Y(s),s≤t).tTheobservationY(t)attimetistheresultofadeterministictransformationofthesignalprocessX(s),s≤t,typicallyalineartransformation,towhicharandomnoiseisadded.Thefilteringproblemistofindthe“best”estimateπX(t)ofthesignalX(t)onthebasisofalltheobservationsY(s),s≤t,orFY.The“best”isunderstoodinthesenseofthesmallestestimationerrortE(X(t)−Z(t))2|FY,whenZ(t)variesoverallFY-measurableprocesses.ttDenoteforanadaptedprocessh(t)π(h)=E(h(t)|FY).(14.1)ttThenbyTheorem2.26(seealsoExercise14.2)thefilteringproblemisinfindingπt(X).375 376CHAPTER14.APPLICATIONSINENGINEERINGANDPHYSICSTwomainresultsofstochasticcalculusareusedtosolvethefilteringprob-lem,Levy’scharacterizationofBrownianmotionandthepredictablerepre-sentationpropertyofBrownianfiltration.GeneralNon-linearFilteringModelLet(Ω,F,P)beacompleteprobabilityspace,andlet(Ft),0≤t≤T,beanon-decreasingfamilyofrightcontinuousσ-algebrasofF,satisfyingtheusualconditions,andsupportingadaptedprocessesX(t)andY(t).Weaimatfindingπt(h)foranadaptedprocessh(t).Inparticular,wecanfindE(g(X(t))|FY),forafunctionofarealvariableg.Whengrangesoveratsetoftestfunctions,theconditionaldistributionofX(t)givenFYisobtained.tMoreoverbytakingg(x)=x,weobtainthebestestimatorπX(t).Weassumethattheprocessh(t),whichwewanttofilterandtheobservationprocessY(t)satisfytheSDEsdh(t)=H(t)dt+dM(t),dY(t)=A(t)dt+B(Y(t))dW(t),(14.2)where:•TheprocessM(t)isaFt-martingale,•TheprocessW(t)isaFt-Brownianmotion,•TheprocessesH(t)andA(t)arerandom,satisfyingwithprobabilityoneTt|H(t)|dt<∞,|A(t)|dt<∞,002T2T2•supt≤TE(h(t))<∞,0EH(t)dt<∞,0EA(t)dt<∞,•ThediffusioncoefficientoftheobservationprocessY(t)isafunctionB(y)ofY(t)only,andnotX.B2(y)≥C>0,andB2satisfiesLipschitzandthelineargrowthconditions.Theorem14.1Foreacht,0≤t≤T,πt(hA)−πt(h)πt(A)dπt(h)=πt(H)dt+(πt(D)+)dW(t),(14.3)B(Y(t))wheredW(t)=dY(t)−πt(A)isanFY-BrownianmotionandD(t)=dM,W(t).B(Y(t))tdtW(t)iscalledtheinnovationprocess.Weoutlinethemainideasoftheproof.By(14.2)th(t)=h(0)+H(s)ds+M(t)and(14.4)0)*tE(h(t)|FY)=E(h(0)|FY)+EH(s)ds|FYds+E(M(t)|FY).tttt0 14.1.FILTERING377ThedifficultyincalculationoftheconditionalexpectationsgivenFYisthatttheprocesses,aswellastheσ-fields,bothdependont.Theorem14.2LetA(t)beanFt-adaptedprocesssuchthatTtE(|A(t)|)dt<∞,andV(t)=A(s)ds.Then00t)t'*t'YYπt(V)−πs(A)ds=EA(s)ds'Ft−E(A(s)|Fs)ds000isaFY-martingale.tProof:Letτ≤TbeanFY-stoppingtime.ByTheorem7.17itisenoughtτtoshowthatE(πτ(V))=E(0πs(A)ds).E(πτ(V))=E(V(τ))bythelawofdoubleexpectationτT=E(A(s)ds)=E(I(s≤τ)A(s))ds00T=E(I(s≤τ)π(A))dssinceI(s≤τ)isFY-measurabless0τ=E(πs(A)ds).0
NotethatbytheconditionalversionofFubini’stheorem,)t'*t'EA(s)ds'G=E(A(s)|G)ds.(14.5)00VerificationofthenextresultisstraightforwardandisleftasanExercise14.4.Theorem14.3LetM(t)beaF-martingale.Thenπ(M)isanFY-martingale.tttCorollary14.4tπt(h)=πt(0)+πs(H)ds+M1(t)+M2(t)+M3(t),(14.6)0whereM(t)aremartingalesnullatzero;M(t)=E(h(X(0))|FY)−h(0),i1ttYtYM2(t)=E0H(s)ds|Ftds−0πs(H)ds,M3(t)=E(M(t)|Ft).Proof:Using(14.4),thefirsttermE(h(X(0))|FY)isa(Doob-Levy)mar-ttingale.ThesecondtermisamartingalebyTheorem14.2,thethirdbytheaboveTheorem14.3.
WewanttousearepresentationofFYmartingales.Thisisdonebyusingthetinnovationprocess. 378CHAPTER14.APPLICATIONSINENGINEERINGANDPHYSICStdY(s)−πs(A)dsYTheorem14.5TheinnovationprocessW(t)=0B(Y(s))isanFt-Brownianmotion.Moreover,dY(t)=πA(t)dt+B(Y(t))dW(t).(14.7)Proof:Byusing(14.2)tYA(s)−E(A(s)|Fs)W(t)=ds+W(t).(14.8)0B(Y(s))Fort>t,writeEW(t)−W(t)|FY=EW(t)−W(t)|FYtttY'A(s)−E(A(s)|Fs)'Y+E'Ftds.tB(Y(s))Therhsiszero,thefirsttermbyTheorem14.3,andthesecondbythecon-ditionalversionofFubini’stheorem.ThusW(t)isanFY-martingale.Itistclearlycontinuous.Itfollowsfrom(14.8)that[W,W](t)=[W,W](t)=t.ByLevy’scharacterizationTheoremtheclaimfollows.
Now,ifFW=FY,(14.9)tthenconditionalexpectationsgivenFYarethesamegivenFW,andTheoremtt8.35onrepresentationofmartingaleswithrespecttoaBrownianfiltrationcanbeusedforthemartingalesMi(t)in(14.6).Commentthatfor(14.9)toholditissufficientthattheSDEforY(t)(14.7)hasuniqueweaksolution.Theorem8.35anditscorollary(8.67)givethattherearepredictableprocessesgi(s),suchthat45TdM,W(t)iMi(t)=gi(s)dW(s),withgi(t)=.0dtItispossibletoshowthatπt(hA)−πt(h)πt(A)g1(t)+g2(t)+g3(t)=πt(D)+,B(Y(t))andtheresultfollowsfrom(14.6).FordetailsseeLiptserandShiryaev(2001)p.319-325. 14.1.FILTERING379FilteringofDiffusionsLet(X(t),Y(t)),0≤t≤T,beadiffusionprocesswithwithrespecttoindependentBrownianmotionsWi(t),i=1,2;Ft=σ(W1(s),W2(s),s≤t),anddX(t)=a(X(t))dt+b(X(t))dW1(t)dY(t)=A(X(t))dt+B(Y(t))dW2(t).(14.10)AssumethatthecoefficientssatisfytheLipschitzcondition,foranyofthefunctionsa,A,b,B,e.g.|a(x)−a(x)|≤K|x−x|;andB2(y)≥C>0.Leth=h(X(t)).Thefunctionhisassumedtobetwicecontinuouslydifferentiable.WeapplyTheorem14.1toh(X(t)).ByItˆo’sformula,Theorem6.1,wehavetth(X(t))=h(X(0))+Lh(X(s))ds+h(X(s))b(X(s))dW(s),100where12Lh(x)=h(x)a(x)+h(x)b(x).2ByTheorem14.1ttπs(Ah)−πs(A)πs(h)πt(h)=π0(h)+πs(Lh)ds+dW(s),(14.11)00B(Y(s))wheretdY(s)−πs(A)dsW(t)=.0B(Y(s))Thelinearcasecanbesolvedandisgiveninthenextsection.Kalman-BucyFilterAssumethatthesignalandtheobservationprocessessatisfylinearSDEswithtimetime-dependentnon-randomcoefficientsdX(t)=a(t)X(t)dt+b(t)dW1(t),(14.12)dY(t)=A(t)X(t)dt+B(t)dW2(t),(14.13)withtwoindependentBrownianmotions(W1,W2),andinitialconditionsX(0),Y(0).DuetolinearitythiscaseadmitsaclosedformsolutionfortheprocessesXandYandalsofortheoptimalestimatorofX(t)givenFY.InthelineartcaseitiseasytosolvetheaboveSDEsandverifythattheprocessesX(t),Y(t)arejointlyGaussian.ItisconvenientinthiscasetousenotationXA(t)=π(X)=E(X(t)|FY).tt 380CHAPTER14.APPLICATIONSINENGINEERINGANDPHYSICSTheorem14.6SupposethatthesignalX(t)andtheobservationY(t)aregivenby(14.12)and(14.13).ThenthebestestimatorXA(t)=E(X(t)|FY)tsatisfiesthefollowingSDE)*A2(t)A(t)dXA(t)=a(t)−v(t)XA(t)dt+v(t)dY(t),(14.14)B2(t)B2(t)wherev(t)=EX(t)−XA(t))2isthesquaredestimationerror.ItsatisfiestheRiccatiordinarydifferentialequationdv(t)A2(t)v2(t)=2a(t)v(t)+b2(t)−,(14.15)dtB2(t)2Cov(X(0),Y(0))withinitialconditionsX(0)andv(0)=Var(X(0))−.var(Y(0))Proof:ApplyTheorem14.1withh(x)=x(seealsoequation(14.11))tohaveA(t)22dXA(t)=a(t)XA(s)dt+πt(h)−πt(h)dY(t)−A(t)XA(t)dt.(14.16)B2(t)Notethat2π(h2)−π(h)=E(X(t)−XA(t))2|FY.tttBecausetheprocessesXandYarejointlyGaussian,XA(t)istheorthogonalprojection,andX(t)−XA(t)isorthogonal(uncorrelated)toY(s),s≤t.ButifGaussianrandomvariablesareuncorrelated,theyareindependent(seeTheo-rem2.19).ThusX(t)−XA(t)isindependentofFYandwehavethatv(t)istdeterministic,v(t)=E(X(t)−XA(t))2|FY=EX(t)−XA(t))2.tTheinitialvaluev(0)isobtainedbytheTheorem2.25onNormalcorrelation,asstatedintheTheorem.Letδ(t)=X(t)−XA(t).Thenv(t)=E(δ2(t)).TheSDEforδ2(t)isobtainedfromSDEs(14.12)and(14.16)asfollowsA2(t)v(t)A(t)v(t)dδ(t)=a(t)δ(t)dt+b(t)dW1(t)−δ(t)dt−dW2(t).B2(t)B(t)ApplyingnowItˆo’sformulatoδ2(t),wefind2A2(t)v(t)22A2(t)v2(t)dδ(t)=2a(t)−δ(t)+b(t)+dtB2(t)B2(t)A(t)v(t)+2δ(t)b(t)dW1(t)−dW2(t).(14.17)B(t) 14.1.FILTERING381Writing(14.17)inintegralformandtakingexpectationt222A(s)v(s)2A(s)v(s)v(t)=v(0)+2a(s)−v(s)+b(s)+ds0B2(s)B2(s)t222A(s)v(s)=v(0)+2a(s)+b(s)−ds,(14.18)0B2(s)whichestablishes(14.15).
Themulti-dimensionalcaseissimilar,withtheonlydifferencebeingthattheequation(14.15)isthematrixRiccatiequationfortheestimationerrorcovariancematrix.TheKalman-Bucyfilterallowson-lineimplementationoftheaboveequa-tions,whichareusedrecursivelytocomputeXˆ(t+∆t)fromthepreviousvaluesofXˆ(t)andv(t).Example14.1:(Modelwithconstantcoefficients)Considerthecaseofconstantcoefficients,dX(t)=aX(t)dt+dW1(t),dY(t)=cX(t)dt+dW2(t).InthiscasedXA(t)=a−v(t)c2XA(t)dt+cv(t)dY(t),(14.19)andv(t)satisfiestheRiccatiequationdv(t)22=2av(t)+1−cv(t).(14.20)dtThisequationhasanexplicitsolution:λtγαe+βv(t)=,(14.21)γeλt+122whereαandβaretherootsof1+2ax−cx,assumedtobeα>0,β<0,2222λ=c(α−β),andγ=(σ−β)/(α−σ),withσ=Var(X(0)).Usingv(t)abovetheoptimalestimatorXA(t)isfoundfrom(14.19).Formoregeneralresultssee,forexample,LiptserandShiryayev(2001),RogersandWilliams(1990),Oksendal(1995).Thereisavastamountofliteratureonfiltering,see,forexample,KallianpurG.(1980),KrishnanV.(1984)andreferencestherein. 382CHAPTER14.APPLICATIONSINENGINEERINGANDPHYSICS14.2RandomOscillatorsSecondorderdifferentialequationsx¨+h(x,x˙)=0.areusedtodescribevarietyofphysicalphenomena,andoscillationsareoneofthem.Example14.2:(Harmonicoscillator)Theautonomousvibratingsystemisgovernedby¨x+x=0,x(0)=0,˙x(0)=1,wherex(t)denotesthedisplacementfromthestaticequilibriumposition.Ithasthesolutionx(t)=sin(t).Thetrajectoriesofthissysteminthephasespacex1=x,x2=˙xareclosedcircles.Example14.3:(Pendulum)Theundampenedpendulumisgovernedby¨x+asinx=0,wherex(t)denotestheangulardisplacementfromtheequilibrium.Itssolutioncannotbeobtainedinterms2ofelementaryfunctions,butcanbegiveninthephaseplane,(˙x)=2acosx+C.Example14.4:(VanderPoloscillator)Insomesystemsthelargeoscillationsaredampened,whereassmallonesareboosted(negativedamping).SuchmotionisgovernedbytheVanderPolequation2x¨−a(1−x)˙x+x=0,(14.22)wherea>0.Itssolutioncannotbeobtainedintermsofelementaryfunctions,eveninthephaseplane.Example14.5:(Rayleighoscillator)2x¨−a(1−x˙)˙x+x=0,(14.23)wherea>0.WeconsiderrandomexcitationsofsuchsystemsbyWhitenoise(inapplied2language)oftheformi=1fi(x,x˙)W˙i(t),whereW˙i(t),i=1,2,areWhitenoiseswithdelta-typecorrelationfunctionsEW˙i(t)W˙j(t+τ)=2πKijδ(τ)dt.Thustherandomlyperturbedequationhastheform
X¨+h(x,X˙)=fi(X,X˙)W˙i(t).iThewhitenoiseisformallythederivativeofBrownianmotion.SinceBrownianmotionisnowheredifferentiable,theabovesystemhasonlyaformalmeaning. 14.2.RANDOMOSCILLATORS383ArigorousmeaningtosuchequationsisgivenbyasystemoffirstorderItˆostochasticequations(asinglevectorvaluedequation).Therepresentationasasystemoftwofirstorderequationsfollowsthesameideaasinthedeterministiccasebylettingx1=xandx2=˙x.UsingTheorem5.20onconversionofStratanovichSDEsintoItˆoSDEs,weendupwiththefollowingItˆosystemofstochasticdifferentialequationsdX1=X2dt,
∂fdX=−h(X,X)+πKf(X,X)k(X,X)dt+212jkj1212∂x2j,k
2fi(X1,X2)dWi(t).i=0Theabovesystemisatwo-dimensionaldiffusion,andithasthesamegeneratorasthesystembelowdrivenbyasingleBrownianmotionB(t),dX1=X2dt,dX2=A(X1,X2)dt+G(X1,X2)dB(t),(14.24)where
∂fk(x1,x2)A(x1,x2)=−h(x1,x2)+πKjkfj(x1,x2),∂x2j,k1/2
G(x,x)=(2π)1/2Kf(x,x)f(x,x).12jkj12k12j,kThesystem(14.24)isarigorousmathematicalmodelofrandomoscillators,andthisformisthestartingpointofouranalysis.Solutionstotheresult-ingFokker-Planckequationsgivedensitiesofinvariantmeasuresorstation-arydistributionsforsuchrandomsystems.ThecorrespondingFokker-Planckequationhastheform∂∂1∂2xp+(Ap)−G2p=0,2ss2s∂x1∂x22∂x2whereps(x1,x2)isthedensityoftheinvariantmeasure.WhenasolutiontotheFokker-Planckequationhasafiniteintegral,thenitisastationarydistributionoftheprocess.However,theymaynotexist,especiallyinsystemswithtrajectoriesapproachinginfinity.Insuchcasesinvariantmeasureswhicharenotprobabilitydistributionsmayexistandprovideinformationabouttheunderlyingdynamics. 384CHAPTER14.APPLICATIONSINENGINEERINGANDPHYSICSToillustrate,considerastochasticdifferentialequationsofordertwowithnonoisein˙xX¨+h(X,X˙)=σB.˙ThecorrespondingItˆosystemisgivenbydX1=X2dt,dX2=−h(X1,X2)dt+σdB(t).InthenotationofSection6.10,thedimensionn=2andthereisonlyoneBrownianmotion,d=1,sothatX(t)=(X(t),X(t))T,b(X(t))=12(b(X(t)),b(X(t)))T=(X(t),−h(X(t),X(t)))Tσ(X(t))=(0,σ)T.The12212diffusionmatrix"#T00a=σσ=2,0σhencethegeneratorisgivenby
2∂1
2
2∂2L=bi+aij∂xi2∂xi∂xji=1i=1j=1∂∂1∂22=x2−h(x1,x2)+σ2.(14.25)∂x1∂x22∂x2AnimportantexampleisprovidedbylinearequationsX¨+aX˙+bX=σB,˙(14.26)withconstantcoefficients.Solutionstotheseequationscanbewritteninageneralformula.)t*X(t)=(exp(Ft))X(0)+(exp(−Fs))(0,σ)TdB(s),0"#01whereF=andexp(Ft)standsforthematrixexponential(see,−b−aforexample,Gard(1988)).SolutionstotheFokker-Plankequationcorrespondingtosomelinearsys-tems(14.26)aregiveninBezenandKlebaner(1996).Non-linearSystemsForsomesystemstheFokker-Plankequationcanbesolvedbythemethodofdetailedbalance.Wegiveexamplesofsuchsystems,fordetailsseeBezenandKlebaner(1996).InwhatfollowsK0,K1denotescalingconstants. 14.2.RANDOMOSCILLATORS385DuffingEquationConsideradditivestochasticperturbationsoftheDuffingequation:X¨+aX˙+X+bX3=W.˙)*a2142ps(x1,x2)=exp−(x2+bx1+x1).2πK02TypicalphaseportraitsanddensitiesofinvariantmeasuresareshowninFigure14.1.RandomOscillatorConsiderthefollowingoscillatorwithadditivenoiseX¨−a(1−X2−X˙2)X˙+X=W.˙Thedensityoftheinvariantmeasureisgivenby)*a(x2+x2)212ps=exp−.4πK0Itfollowsthatwhena=0thesurfacerepresentingtheinvariantdensityisaplane.Whena>0afourthordersurfacewithrespecttox1,x2hasmaximuminacurverepresentingthelimitcycleofthedeterministicequation.ConsidernowthesameequationwithparametricnoiseoftheformX¨−a(1−X2−X˙2)X˙+X=W˙+(X2+X˙2)W˙.01Thedensityoftheinvariantmeasureisgivenby√K0(a+2πK1)2244−ps=(K0+2K1x1x2+K1x1+K1x2)4√22K1(x+x)2aarctan√12exp√K0.4πK0K1TypicalphaseportraitsanddensitiesoftheinvariantmeasuresareshowninFigure14.2.ASystemwithaCylindricPhasePlaneConsiderrandomperturbationstoasystemwithacylindricphase−π≤x<π,−∞x)=P(B(t)>x)+P(B(t)<−x)=2P(B(t)>x)=P(M(t)>x).∞|x|∞3x2Exercise3.10:ByTheorem3.18,E(Tr)=trf(t)dt=√tr−2e−2tdtx02π0|x|∞1x2=√s−r−2e−2sds.Theintegralconvergesatinfinityforanyr.Atzero2π0itconvergesonlyforr+1<1.2,∞y(2y−x)−(2y−x)22Exercise3.11:fM(y)=fB,M(x,y)dx=3/2e2tdx,−∞,−∞πty−(2y−x)2−y2212=1/2de2t=e2t=2fB(y),by(3.16).πt−∞πt391 392SOLUTIONSTOSELECTEDEXERCISESExercise3.12:mins≤tB(s)=−maxs≤t−B(s).LetW(t)=−B(t),thenitisalsoaBrownianmotionandwehaveP(B(t)≥x,mins≤tB(s)≤y)=−2y+xP(W(t)≤−x,maxs≤tW(s)≥−y)=1−Φ(√).Withφ=Φ,fort2−2y+xy≤0,x≥y,f(x,y)=−∂P(B(t)≥x,m(t)≤y)=2φ(√)=B,m,∂x∂ytt2221−(−2y+x)−2y+x2(x−2y)−(2y−x)√e2t√=e2t.t2πtπt3/2Exercise3.13:Leta>0.LetD={(x,y):y−x>a}.Thenwehave∞y−aP(M(t)−B,(t)>a)=DfB,M(x,y)dxdy,=0−∞fB,M(x,y)dxdy∞y−a(2y−x)−(2y−x)2∞y−a−(2y−x)222∂=3/2e2tdxdy=(−(e2t)dx)dy=,0−∞πttπ0−∞∂x∞−(y+a)2∞−(y+a)2∞−u2211e2tdy=2√e2tdy=2√e2tdutπ002πta2πt=2P(B(t)>a).Exercise3.14:T2=inf{t>0:B(t)=0}.Sinceanyinterval(0, )containsazeroofBrownianmotion,T2=0,thesecondzeroisalsozero.Exercise3.15:Bytheaboveargument,anyzeroofBrownianmotionisalimitfromtherightofotherzeroes.BythedefinitionofT,itisazeroofX,butisnotalimitofotherzeroes.ThusXisnotaBrownianmotion.tB(1/t)tB(1/t)B(1/t)Exercise3.16:limsupt→∞√=1.√=√.2tlnlnt2tlnlnt2(1/t)ln(−ln(1/t))√B(τ)Thuswithτ=1/t,limsupτ→0=1.Similarforliminf.2τln(−lnτ)Exercise3.17:(B(e2αt1),B(e2αt2),...,B(e2αtn))isaGaussianvector.Thefinite-dimensionaldistributionsofX(t)=e−αtB(e2αt)areobtainedbymul-tiplyingthisvectorbyanon-randomdiagonalmatrixA,withthediagonalelements(e−αt1,e−αt2,...,e−αtn).Thereforefinite-dimensionaldistributionsofXaremultivariateNormal.Themeaniszero.Lets0.HencetTn0X(s)dB(s)=0X(s)I(0,t](s)dB(s)=i=0ξi(B(ti+1∧t)−B(ti∧t)),whereti∧t=min(ti,t).TheItˆointegraliscontinuousasasumofcontinuousfunctions.Exercise4.3:Thefirststatementfollowsbyusingmomentgeneratingfunc-22µnt+σt/2222tionsenconvergesimpliesthatµn→µandσn→σ≥0.Ifσ=0,thenthelimitisaconstantµ,otherwise,thelimitisN(µ,σ2).AnItˆointegralofanonrandomfunctionisalimitofapproximatingItˆoin-tegralsofsimplenonrandomfunctionsXn(t).SinceXn(t)takesfinitelymanyTnonrandomvalues,Xn(t)dB(t)hasaNormaldistributionwithmeanzero0T2Tandvariance0Xn(t)dt.Bythefirststatement0X(t)dB(t)hasNormalTdistributionwithmeanzeroandvarianceX2(t)dt.Analternativederiva-0tionofthisfactisdonebyusingthemartingaleexponentialofthemartingaletuX(s)dB(s).0Exercise4.5:M(T)hasaNormaldistribution,EM2(T)<∞.ByJensen’sinequalityforconditionalexpectation(p.45)withg(x)=x2,E(M2(T)|F)≥(E(M(T)|F))2=M2(t).ThereforeE(M2(t))≤E(M2(T))ttandM(t)isasquare-integrablemartingale.ThecovariancebetweenM(s)andM(t)−M(s)fors0)+1I(B(T)=0).Themartingale2propertyholdsalsofort=Tbydominatedconvergence. SOLUTIONSTOSELECTEDEXERCISES395Exercise4.18:dX(t)=tdB(t)+B(t)dt,d[X,X](t)=(dX(t))2=t2dt,t[X,X](t)=s2ds=t3/3.0ttttExercise4.19:X(t)=tB(t)−sdB(s)=sdB(s)+B(s)ds−sdB(s)0000t=B(s)ds.ThusX(t)isdifferentiableandoffinitevariation.Hence0t[X,X](t)=0.Itˆointegralsoftheformh(s)dB(s)haveapositivequadratic0tvariation.InthiscasetheItˆointegralisoftheformh(t,s)dB(s).0ExercisestoChapter5tExercise5.1:ByExample4.5,b(s)dB(s)isaGaussianprocess.0tttX(t)=a(s)ds+b(s)dB(s)isalsoGaussianasa(s)dsisnon-random.000Exercise5.3:dX(t)=X(t)(B(t)dt+B(t)dB(t)).X(t)=E(R)(t),wherettt12t(B(s)−B(s))ds+B(s)dB(s)R(t)=B(s)ds+B(s)dB(s).ThusX(t)=e020.00Exercise5.4:LetdM(t)=B(t)dB(t).ThendX(t)=X(t)dt+dM(t),whichisaLangevintypeSDE.SolvingsimilarlytoExample5.6weobtainX(t)=e−t(1+tesdM(s))=e−t(1+tesB(s)dB(s)).00TheSDEisnotofthediffusiontypeasσ(t)=B(t).ByintroducingY(t)=B(t),itisadiffusionintwodimensions.Exercise5.5:LetU(t)=E(B)(t).ThendU(t)=U(t)dB(t).dU2(t)=2U(t)dU(t)+d[U,U](t)=2U2(t)dB(t)+U2(t)dt=U2(t)(2B(t)+dt).SothatU2(t)=E(2B(t)+t).Exercise5.6:dX(t)=X(t)(X(t)dt+dB(t)).IfdY(t)=X(t)dt+dB(t)thenX(t)=E(Y)(t).Exercise5.10:Bydefinition,P(y,t,x,s)=P(B(t)+t≤y|B(s)+s=x)=P(B(t)−B(s)+t−s≤y−x|B(s)+s=x)=P(B(t)−B(s)+t−s≤y−x)byindependenceofincrements.B(t)−B(s)hasN(0,t−s)distribution,andy−x−t+sP(y,t,x,s)=Φ√.t−stExercise5.12:X(t)=X(s)+1dB(s).Assumingitisamartingale022ttEX(t)=0,EX(t)=EX(s)+1dB(s)=E(X(s)+1)ds=t.00deuX(t)=ueuX(t)dX(t)+1u2euX(t)d[X,X](t).d[X,X](t)=(X(t)+1)dt,and2deuX(t)=ueuX(t)X(t)+1dB(t)+1u2euX(t)(X(t)+1)dt.Takingexpecta-2tionm(t)=1+1u2EteuX(s)(X(s)+1)ds=1+1u2tEeuX(s)X(s)ds+202012tuX(s)∂mu2uX(t)u2uX(t)uEeds.Thus=EeX(t)+Eeandthede-20∂t22siredPDEfollows. 396SOLUTIONSTOSELECTEDEXERCISESExercisestoChapter61∂221Exercise6.1:DenoteL=andf(x,t)=eux−ut/2.ThenLf=u2f2∂x2and∂f=−1u2f.ThusLf+∂f=0.ByIto’sformulaandCorollary6.4∂t2∂tuB(t)−u2t/2∂ff(B(t),t)=eisamartingale.SincefsolvesLf+=0forany∂t∂ffixedu,takepartialderivativeandinterchangetheorderofdifferentiation∂utohavethatforanyfixedu,∂Lf+∂∂f=L(∂f)+∂∂f=0,sothat∂u∂u∂t∂u∂t∂u∂fsolvesthebackwardequationforanyfixeduandinparticularforu=0.∂uCalculatingthederivativeswegettheresult.22Exercise6.2:Thegeneratorisgivenby(6.30)L=σd−αd.The2dx2dx2∂2f∂f∂fbackwardequationisσ−α=.Thefundamentalsolutionisgiven2∂x2∂x∂tbytheprobabilitydensityofthesolutiontotheSDEX(t).p(t,x,y)=∂P(t,x,y),whereP(t,x,y)=P(X(t)≤y|X(0)=x).By(5.13)∂yP(t,x,y)=P(xe−αt+e−αtteαsdB(s)≤y)=P(teαsdB(s)≤yeαt−x)√00α(yeαt−x)tαs1αt=Φ√sinceedB(s)hasN(0,(e−1))distribution.Thuseαt−10α√√α(yeαt−x)αt∂αep(t,x,y)=P(t,x,y)=φ√√withφdenotingthedensity∂yeαt−1eαt−1ofN(0,1).2Exercise6.4:L=1∂+cx∂.Takef(x)=x2,thenLf(x)=1+2cx2and2∂x2∂xbyTheorem6.11X2(t)−t(Lf+∂f)(X(s))ds=X2(t)−t−2ctX2(s)ds0∂t0isamartingale.Exercise6.6:UsingCorollary6.4orItˆo’sformula,wehavedf(B(t)+t)=f(B(t)+t)dB(t)+f(B(t)+t)dt+1f(B(t)+t)dt.Anecessarycondition2forf(B(t)+t)tobeamartingaleisthatdttermiszero.Thisgivesf(x)+1f(x)=0.Forexample,takef(x)=e−2xandcheckdirectlythate−2(B(t)+t)2isamartingale.(e−2(B(t)+t)=E(2B)(t)).2∂f∂f∂fExercise6.7:df(X(t),t)=∂xdX(t)+∂x2d[X,X](t)+∂tdt.Theterm∂fwithdB(t)isgivenbyσ(X(t),t)(X(t),t)dB(t).ThusthePDEforfis∂x∂fσ(x,t)(x,t)=1.∂xExercise6.8:Bylettingv=yweobtainafirstorderdifferentialequationy+2µy+2=0.UsetheintegratingfactorV(x)=expx2µ(s)ds,toσ2σ2aσ2(s)obtain(yV)=−2Vandy=−12V.Theresultfollows.σ2Vσ2x2µ2µ−ds−xExercise6.9:Thescalefunction(6.50)S(x)=eσ2=eσ2gives2µ2µ2µ2µ2µ−xS(x)−S(a)−x−a−b−aS(x)=Ceσ2.Px(Tb0thenexp(−x3−2α/σ2)xexp(y3−2α/σ2)dy∼Cσ23−2α1σ2y2αx2asx→∞and∞exp(−x3−2α/σ2)xexp(y3−2α/σ2)dydxconverges.Conse-11σ2y2αquently,theprocessexplodes.When3−2α<0thentheintegraldivergesandthereisnoexplosion.Thecaseα=3/2needsfurtheranalysis.Exercise6.13:CheckconditionsofTheorem6.28.µ(x)=0,σ(x)=1.I1=x0∞−∞1du=∞,I2=x01du=∞.HenceB(t)isrecurrent.Fortheprocess∞B(t)+t,µ(x)=1andσ(x)=1.I=exp−2(u−x)du=1<∞.Thus2x002B(t)+tistransient.TheOrnstein-Uhlenbeckprocessislefttothereader.Exercise6.15:Forn=2,S(x)=lnx.Sothatfor00.Soif0<2ba/σ2<1,0isaregularboundarypoint,andifba<0,0isanabsorbingboundary.ExercisestoChapter7Exercise7.1:Gt⊆Ft.Letsq.T2Exercise7.12:sign(B(s))ds=T<∞.ThusX(t)isamartingale.0t2[X,X](t)=sign(B(s))ds=t.ByLevy’stheoremXisaBrownianmotion.0tExercise7.13:[M,M](t)=e2sds=1(e2t−1).Itsinversefunctionis02g(t)=1ln(2t+1).M(g(t))isaBrownianmotionbytheDDSTheorem7.37.2ttExercise7.16:X(t)=X(0)+A(t)+M(t)=X(0)+µ(s)ds+σ(s)dBs00isalocalmartingale.ThereforeA(t)=X(t)−M(t)isalocalmartingaleasadifferenceoftwolocalmartingales.A(t)iscontinuous.ByusingCorollary7.30,A(t)hasinfinitevariationunlessitisaconstant.SinceA(t)isoffinitevariationitmustbezero.Theresultfollows.Exercise7.17:ByItˆo’sformulat∂f∂2ft∂ff(B(t),t)−0∂t(B(s),s)+∂x2(B(s),s)ds=0∂x(B(s),s)dB(s).Therhs.isacontinuouslocalmartingale,andthelhs.isoffinitevariation.Thiscan∂fonlyhappenwhenthelocalmartingaleisaconstant.Thus=0,andfis∂xaconstantinx,henceafunctionoftalone.Exercise7.18:WeknowY(t)=1B2(t)−1t,andB2(t)=2Y(t)+t.Thus22B(t)=sign(B(t))2Y(t)+t.ThereforedY(t)=sign(B(t))2Y(t)+tdB(t)ordY(t)=2Y(t)+tdW(t),wheredW(t)=sign(B(t))dB(t),WisaBrow-nianmotion.WeakUniquenessfollowsbyTheorem5.11.Alternatively,onecanproveitdirectlybycalculatingofmomentsofY(t),µ(t)=EYn(t)(bynn(n−1)tusingIto’sformulaµn(t)=20(sµn−2(s)+2µn−1(s))ds)andcheckingCarleman’scondition(µ)−1/n=∞,seeFeller(1971).n SOLUTIONSTOSELECTEDEXERCISES399ExercisestoChapter8Exercise8.1:I(τ1,τ2](t)=I[0,τ2](t)−I[0,τ1](t)isasumoftwoleft-continuousadaptedfunctions.Letτbeastoppingtime.ThenX(t)=I[0,τ](t)isadapted.∞{X(t)=0}={t>τ}=n=1{τ≤t−1/n}∈Ft,since{τ≤t−1/n}∈Ft−1/n⊂Ft.Seep.54.Exercise8.2:Theleft-continuousmodificationoftheprocessH(t−δ+)isadaptedandas→0tendstoH(t−δ).tExercise8.4:LetX(t)=H(s)dM(s),thenitisalocalmartingaleasa0stochasticintegralwithrespecttoalocalmartingale.t[X,X](t)=H2(s)d[M,M](s).ThusE[X,X](T)<∞.ByTheorem07.35.thisconditionimpliesXisasquareintegrablemartingale.11Exercise8.5:E(N(t−)dM(t))=0,sothatVar(N(t−)dM(t))=00E(1N(t−)dM(t))2=E1N2(t−)d[M,M](t).[M,M](t)=[N,N](t)=00N(t),sothatVar(1N(t−)dM(t))=E1N2(t−)dN(t)=E(N2(τ)),00τi≤1i−1whereτidenotesthetimeofthei-thjumpofN.ButN(τi)=i,sothat12N(1)2N(1)−12Var(N(t−)dM(t))=E((i−1)=E((i−1))=E(k)0τi≤1i=1k=0=E(2N3(1)−3N2(1)+N(1))/6,sinceτ≤1isequivalenttoi≤N(1),andin23212k=(2(n+1)−3(n+1)+n+1)/6.AlternativelyN(t−)dN(t)k=00canbeobtainedbyusingformula(1.20)p.12.Computemomentsusingthesmgfm(s)=EesN(1)=e−1ee.m(0)=EN(1)=1,m(0)=EN2(1)=2,1m(3)(0)=EN3(1)=5,givingVar(N(t−)dM(t))=5/6.0Exercise8.6:1.S∧TandS∨Tarestoppingtimes,because{S∧T>t}={S>t}∩{T>t}∈Ft,{S∨T≤t}={S≤t}∩{T≤t}∈Ft.2.Theevents{S=T},{S≤T}and{S0andE(Y/EY)=1,letΛ=Y/EYanddefine2dQ/dP=Λ=eX−µ−σ/2.ByTheorem10.4,theQdistributionofYisN(µ+σ2,σ2).ThusEYI(Y>K)=EYEΛI(Y>K)=EYEI(Y>K)=Q22eµ+σ/2Q(Y>K).Finally,EYI(Y>K)=eµ+σ/2Φ((µ+σ2/2−K)/σ),using1−Φ(x)=Φ(−x).tExercise10.4:X(t)=B(t)+cossds.ThusbyGirsanov’sTheorem10.160T1T2−cossdB(s)−cossdsdQ/dP=e020.X(T)−X(0)−1[X,X](T)Exercise10.5:WithX=H·B,E(X)(T)=e2.[H·B,H·B](T)=d[Hi·Bi,Hj·Bj]=dTHi(t)Hj(t)d[Bi,Bj](t)i,j=1i,j=10dT(Hi(t))2dt=T|H(s)|2ds,since[Bi,Bj](t)=0fori=j.Thusi=100nTii1nTi2H(s)dB(s)−(H(s))dsE(H·B)(T)=ei=102i=10T1T2H(s)dB(s)−|H(s)|dse020.T2T2Exercise10.8:SinceH(t)isboundedEH(t)d[N,¯N¯](t)=EH(t)dt00isfiniteandM(t)isamartingale.By(9.5)t6H(s)dN¯(s)−∆(H·N¯)(s)E(M)(t)=e0s≤t(1+∆(H·N¯)(s))e.ButthejumpsoftheintegraloccuratthepointsofjumpsofN¯and∆N¯=∆N,sothat∆(H·N)(s)=H(s)∆N(s).Next,6e−∆(H·N¯)(s)=e−s≤tH(s)∆N(s)=s≤tt6−H(s)dN(s)ln(1+H(s))∆N(s)e0.Proceeding,(1+∆(H·N¯)(s))=es≤t=s≤ttln(1+H(s))dN(s)e0,andtheresultfollows.Exercise10.9:Using(10.52)or(10.51)withµi(x,t)=µix,i=1,2,σ(x,t)=σx,Pcorrespondstoµ1,B(t)isaP-Brownianmotion.2µ2−µ1(µ2−µ1)dQµ2−µ1B(T)−TΛ(X)T==E(B)(T)=eσ2σ2.ReplaceB(T)byitsdPσexpressionasafunctionofX.InthiscaseB(T)=(ln(X(T))−(µ−1σ2)T)/σ.X(0)12Exercise10.11:ByCorollary10.11M(t)isaQ-martingaleifandonlyifM(t)Λ(t)isaP-martingale.d(M(t)Λ(t))=M(t)dΛ(t)+Λ(t)dM(t)+d[M,Λ](t)=M(t)dΛ(t)+Λ(t)dM(t)−Λ(t)dA(t)+d[M,Λ](t).ThefirsttwotermsareP(local)martingales,asstochasticintegralswithrespecttomartingales(Λ(t)isaP-martingale).Thus−Λ(t)dA(t)+d[M,Λ](t)=0.But[A,Λ](t)=0,sinceAiscontinuousandfinitevariation.ThusdA(t)=d[M,Λ](t)/Λ(t). 402SOLUTIONSTOSELECTEDEXERCISESExercise10.13:1.UsingGirsanov’sTheorem10.15,thereexistsanequivalentmeasureQ(definedbyΛ=dQ/dP=e−T/2−B(T))suchthatW(t)=B(t)+tisa1Q1-martingale.WeshowthatunderQ1,N(t)remainstobeaPoissonprocesswithrate1.ThisfollowsbyindependenceofN(t)andB(t)underP,EeuN(t)=E(ΛeuN(t))=E(e−T/2−B(T)euN(t))=E(e−T/2−B(T))E(euN(t))QPPPP=E(euN(t)),sinceEΛ=1.Thisshowsthatone-dimensionaldistributionsPPofN(t)arePoisson(1).Similarly,theincrementsofN(t)underQ1arePoissonandindependentofthepast.Thestatementfollows.2.LetΛ=Λ(T)=dQ2=ΛΛ=e−T−2B(T)e−T+N(T)ln212dP=e−2T−2B(T)+N(T)ln2.ThenB(t)+2tisaQ-Brownianmotion,andN(t)isa2Q-Poissonprocesswithrate2.ToseethisEQ(X)=EP(ΛX)=EP(Λ1Λ2X),foranyXandEPΛ1=EPΛ2=1.SounderQ2,N(t)−2tisamartingale,andB(t)+2tisamartingale,thusX(t)=B(t)+N(t)isaQ2-martingale.3.Theprobabilitymeasuresdefinedasfollowsfora>0areequivalentto22P,dQ=e−aT/2−aB(T)e(1−a)T+N(T)lna=e(1−a−a/2)T−aB(T)+N(T)lnadP,aandmaketheprocessesB(t)+atintoaQa-Brownianmotion,andN(t)intoaQa-Poissonprocesswithratea.ExercisestoChapter11Exercise11.3:IfX=aS1+bforsomeaandbthenEQ(X/r)=aEQ(S1/r)+b=aS0+bthesameforanyQ.NotethatsincethevectorsSandβarenotcollinear,therepresentationofXbyaportfolio(a,b)isunique.Taketheclaimthatpays$1whenthestockgoesupandnothinginanyothercase,thenthisclaimisunattainable.EQ(X)=1.5(0.2+pd)dependsonthechoiceofpd.Exercise11.5:1.EQ(M(t)|Fs)=EP(M(t)|Fs)=M(s)a.s.Takes=0,EPM(t)=EQM(t)=M(0).2.Ifaclaimisattainable,X=V(T),whereV(t)/β(t)isaQ-martingale.ItspriceattimetisC(t)=V(t),whichdoesnotdependonthechoiceofQ.BythemartingalepropertyV(t)=β(t)EQ(V(T)/β(T)|Ft).ThisshowsthatC(t)=β(t)EQ(X/β(T)|Ft)isthesameforallQ’s.Exercise11.10:Considercontinuoustime.d(S(t))=dS(t)+S(t)d(1).β(t)β(t)β(t)dS(t)β(t)S(t)1tβ(u)S(u)tdβ(s)dR(t)==d()−β(t)d().R(t)=d()+.S(t)S(t)β(t)β(t)0S(u)β(u)0β(s)Thefirstintegralisamartingale,asastochasticintegralwithrespecttoaQ-tdβ(s)martingale.ThusEQR(t)=0β(s),whichistherisk-freereturn.IndiscretetimethestatementeasilyfollowsfromthemartingalepropertyofS(t)/β(t).Exercise11.11:LetX(t)=S(t)e−rt.ThenitiseasytoseethatX(t)satisfiestheSDEdX(t)=σX(t)dB(t),foraQ-BrownianmotionB(t).By SOLUTIONSTOSELECTEDEXERCISES403Itˆo’sformulad(1)=−σ1(dB(t)−σdt)=−σ1dW(t),withdW(t)=X(t)X(t)X(t)dB(t)−σdt.W(t)isaQ-Brownianmotionwithdrift.Usingthechange2ofmeasuredQ/dQ=eσB(T)−σT/2,byGirsanov’stheoremW(t)isaQ-11Brownianmotion.Finally,−W(t)isalsoaQ1-Brownianmotion,andwecanremovetheminussignintheSDE.Exercise11.12:BytheTheorem11.7,C(t)=E(S(t)(S−K)+|F)=Q1S(T)TtSE((1−K)+|F).Theconditionaldistributionof1/S(T)givenFundertQ1S(T)ttQisobtainedbyusingtheSDEfor1/S(t).WithY(t)=1/X(t)=ert/S(t),1wehavefromabovedY(t)=σY(t)dW(t)and1/S(t)=Y(t)e−rt.Usingtheproductruled(1)=1(−rdt+σdW(t)).Thus1/S(t)isastochasticS(t)S(t)2exponential,and1/S(T)=(1/S(t))e(T−t)(−r−σ/2)+σ(W(T)−W(t)).ThusgivenFt,theQ1conditionaldistributionof1/S(T)isLognormalwithmeanandvariance−lnS(t)−(T−t)(r+σ2/2)andσ2(T−t).UsingcalculationsfortheE(1−X)+similartop.313,werecovertheBlack-Scholesformula.Exercise11.14:Fromthefirstequationb(t)=(V(t)−a(t)S(t))e−rt.Puttingitintheself-financingconditiondV(t)=a(t)dS(t)+b(t)d(ert),wegettheSDEforV(t).Fortheotherdirection,letb(t)beasabove.ThenV(t)=a(t)S(t)+b(t)ert,moreovertheSDEforV(t)givestheself-financingcondition.Exercise11.17:LetF(y)=e−rTE((S/S−y)+),thenC=SF(K/S).TNow,∂C/∂S=F(K/S)−KF(K/S)/Sand∂C/∂K=F(K/S).ThusS∂C/∂S+K∂C/∂K=SF(K/S)=C.Theexpressionfor∂CintheBlack-∂SScholesmodelfollowsfromtheBlack-Scholesformula.ExercisestoChapter12Exercise12.1:UseTheorem12.5,C(t)=E(β(t)(P(s,T)−K)+|F)Qβ(s)tβ(t)β(t)=EQ(β(s)P(s,T)I(P(s,T)>K)|Ft)−KEQ(β(s)I(P(s,T)>K)|Ft).ForthefirsttermtakeΛ2basedontheQ-martingaleP(s,T)/β(s),s≤T,whichcorrespondstonumeraireP(t,T),orT-forwardmeasure.Sincetheexpecta-P(s,T)tionmustbeunity,Λ2=β(s)P(0,T).(Formula(12.53)).ThenthefirsttermisP(s,T)β(t)EQ(β(s)X|Ft)=P(t,T)EQ2(X|Ft).ForthesecondtermconsiderΛ1=1/(P(0,s)β(s))basedonthemartingaleP(t,s)/β(t),t≤s,whichcorrespondsβttonumeraireP(t,s),ors-forwardmeasure.ThenforanyX,EQ(βsX|Ft)=P(t,s)EQ1(X|Ft).ThisgivesforthesecondtermKP(t,s)Q1(P(s,T)>K|Ft),andtheresultfollows. 404SOLUTIONSTOSELECTEDEXERCISES√Exercise12.3:1.FollowsbyE|X|≤EX2.Tn−12.Byadditivityoftheintegral0i=0H(ti,s)W(ti+1)−W(ti)ds=n−1Ti=00H(ti,s)dsW(ti+1)−W(ti).n−13.LetXn(s)=i=0H(ti,s)W(ti+1)−W(ti).ThenXn(s)isanapprox-T2T2imationtotheItˆointegralX(s)=H(t,s)dW(t).EX(s)=EH(t,s)ds<00T2∞,andunderthestatedconditionsE(Xn(s)−X(s))dsconvergestozero.02TTThisimpliesconvergesinLandinprobabilityof0Xn(s)dsto0X(s)ds.n−1TTherhsi=00H(ti,s)dsW(ti+1)−W(ti)isanapproximationofItˆointegralofY(t),andconvergestoitinprobability.∂lnP(t,T)Exercise12.5:Usetheformulaforthebond(12.32).f(t,T)=−.∂TAcapisasumofcaplets,whicharepricedby(12.64).ttExercise12.7:LetY(t)=γ(s)dB(s)andX(t)=β(s)dB(s)forsome00β(s)tobedetermined.ThenE(X)(t)=c+(1−c)E(Y)(t).ThusdE(X)(t)(1−c)dE(Y)(t)(1−c)E(Y)(t)γ(t)dX(t)===dB(t),andE(X)(t)c+(1−c)E(Y)(t)c+(1−c)E(Y)(t)t1t2γ(s)dB(s)−γ(s)ds(1−c)E(Y)(t)γ(t)(1−c)γ(t)e020β(t)==.ttc+(1−c)E(Y)(t)γ(s)dB(s)−1γ2(s)dsc+(1−c)e020Existenceofforwardratesvolatilitiesfollowsfromequation(12.76).Exercise12.9:AtTithefollowingexchangeismade:theamountreceivedisfi−1(Ti−Ti−1)andpaidoutk(Ti−Ti−1).TheresultingamountattimeTiis1/P(Ti−1,Ti)−1−kδ,using1/P(Ti−1,Ti)=1+fi−1(Ti−Ti−1).Thusthevalueattimetoftheswapisn))**
β(t)1Swap(t,T0,k)=EQ−1−kδ|Ft.β(Ti)P(Ti−1,Ti)i=1Usingthemartingalepropertyofthediscountedbonds(12.4),weobtainthatβ(t)EQβ(Ti)|FTi−1=P(Ti−1,Ti).TheresultisobtainedbyconditioningonFTi−1intheabovesum.Exercise12.10:Followsfromthepreviousexercise.Exercise12.11:Considertheportfolioofbondsthatatanytimett)=P(L1>t,L2>t,...,Lx>t)=(P(L>t))x=(e−a(x)t)x=e−xa(x)t.1tExercise13.7:1.π(t)=I(X(s−)≥k)dNλ(s)hasjumpsofsize10k≥1k1tone.VerifythatAt=0λXsdsisitscompensator.Similarlyfortheprocesstµπ2(t)=0k≥1I(X(s−)≥k)dNk(s).2.X(t)=X(0)+A1(t)−A2(t)+π1(t)−A1(t)−π2(t)+A2(t).ExercisestoChapter14Exercise14.3:E(M(t)|Gs)=E(E(M(t)|Fs)|Gs)=E(M(s)|Gs)=M(s).Exercise14.5:LetGtbethetrivialσ-field,thenWB(t)=E(W(t)|Gt)=0. 406SOLUTIONSTOSELECTEDEXERCISESExercise14.6:UsingformallyTheorem14.6witha(t)=b(t)=0andA(t)=B(t)=1,dXA(t)=−v(t)XA(t)dt+v(t)dY(t),anddv(t)=−v2(t)dt.HencedXA(t)=−(1/t)XA(t)dt+(1/t)dY(t).Weobtainv(t)=1/tandXA(t)=Y(t)/t.ThesecanbeobtaineddirectlyusingY(t)=ct+W(t).Exercise14.8:dX(t)=(µ+σ2/2)X(t)dt+σX(t)dB(t).ApplyTheorem14.6witha(t)=µ+σ2/2,b(t)=σ,A(t)=B(t)=1,seeExample14.1. 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Indexadaptedprocess,49,241chainrule,14admissiblestrategy,293,300changeofdrift,278arbitrage,289changeofmeasure,267,271,279arcsinelaw,75changeofnumeraire,310Asianoptions,315changeoftime,205attainableclaim,293,296changeofvariables,13,220Chapman-Kolmogorovequation,135backwardequation,177characteristicfunction,38Barrieroptions,318coefficientsoftheSDE,126Bayesformula,274compensator,228,229,245,257,BDGinequality,201259,362Besselprocess,175completemarketmodel,296,302bestestimator,46conditionaldistribution,43BGMmodel,341conditionalexpectation,27,43Binomialmodel,291continuityofpaths,48Birth-Deathprocess,356continuous,1bivariatenormal,42continuousmartingales,202Black’sformula,343,348convergencealmostsurely,38Black-Scholes,299,303convergenceinLr,38Black-ScholesPDE,305,326convergenceindistribution,38bond,323convergenceinprobability,38Borelσ-field,29convergenceofexpectations,39bracketprocess,218convergenceofrandomvariables,38branchingdiffusion,351correlatedBrownianmotions,89,branchingprocess,360120,307BrownianBridge,132,205countingprocess,249,252Brownianmotion,56,63,75,132,covariance,40,103,114,120158,165,172,181,278,cross-currencyoptions,315283currencyoptions,312c`adl`ag,4,211Davis’inequality,201calloption,287decompositionofFVfunctions,36cap,348diffusion,302,355caplet,339,340,343413 414INDEXdiffusioncoefficient,116Gaussianprocess,42,103,133,147,diffusionoperator,143330Dirichletclass(D),197genefrequencies,354DiscreteTimeRiskModel,209generator,157,262,263,361distribution,26,31Gronwall’sinequality,18,363dominatedconvergence,46growthconditions,18Doob’sdecomposition,214,245Doob’sinequality,201H¨olderconditions,17Doob-Levymartingale,51,377harmonicfunction,162,177Doob-Meyerdecomposition,214hedging,301drift,303higherorderSDE,179dualpredictableprojection,229hittingtime,69,193,353Dynkin’sformula,149,361hittingtimes,73HJMmodel,331EMMassumption,331Ho-Leemodel,334ergodic,179evanescent,48impliedvolatility,307exittime,69,160,161increment,1exoticoptions,288indicator,30expectation,26,33integrationbyparts,12,112,113,explosion,166,263,369220,252extinctionprobability,353invariantdistribution,170invariantmeasure,385Fatous’lemma,46isometry,94,96,216,234Fellerproperty,168Itˆointegral,91,93,95,103,111,Feynman-Kacformula,155,326202field,22,24Itˆo’sformula,108,111,116,176,filteredprobabilityspace,21,49220,234,235,251filteringequation,376filtration,23,24,240jointdistribution,32finitevariation,4JordanDecomposition,6fluidapproximation,371jump,3Fokker-Plankequation,144Kalman-Bucyfilter,379forwardequation,144forwardmeasure,336Lamperti’sTimeChange,208forwardrate,331Langevinequation,127FractionalBrownianmotion,120LawofIteratedLogarithm,79Fubini’stheorem,53,240LawofLargeNumbers,79fundamentalsolutionofPDE,143Lebesguedecomposition,36FXoptions,312Lebesgueintegral,33,34LebesgueMeasure,30Gambler’sRuin,192Lebesgue-StieltjesIntegral,34Gaussiandistribution,41 INDEX415left-continuous,2PDE,144Levy’scharacterization,203Poissonprocess,86,87,256,368LIBOR,341polarizationidentity,9likelihood,283,284populationdependence,359,367linearSDE,131,384positiverecurrent,179Lipschitzconditions,17predictable,25,83,212,228–230,localmartingale,101,195233,234,241,244localtime,222predictablequadraticvariation,229localization,195,363predictablerepresentation,237Lookbackoptions,317prey-predator,367Lotka-Volterramodel,366price,301pricingbyPDE,305markedpointprocesses,249pricingequation,302Markovprocess,159,262probability,26,29Markovproperty,67probabilitydensity,32,35martingale,50,65,100,177,183productrule,113,220martingaleexponential,82,90progressivelymeasurable,241martingalemeasure,296purejumpprocess,258martingalerepresentation,237putoption,288martingalesquareintegrable,184martingaletransform,84quadraticvariation,8,63,101,109,martingalesinRandomWalk,82198,200,218,219,276,maximumofBrownianmotion,71306meanexittime,161Quantooptions,315measureofjumps,244Merton’smodel,327Radon-Nikodymderivative,271Meyer-Tanakaformula,223randommeasure,244minimumofBrownianmotion,72randomoscillator,385momentgeneratingfunction,38,41randomvariable,23,26,30moments,37randomwalk,81monotoneconvergence,46recurrence,73,81reflectionprinciple,74naturalscale,163regularfunctions,4,211nonlinearfilter,376renewalprocess,255Normaldistribution,41RiemannIntegral,9numeraire,311right-continuous,2,49ruinprobability,193,209occupationtime,222optionpricingformula,311scalefunction,163optional,241SDE,126,173optionalstopping,189self-financingportfolio,293,298Ornstein-Uhlenbeck,127,172,284semimartingale,211,370semimartingaleexponential,224 416INDEXsharpbracket,229,233,235,361strongMarkovProperty,68sigma-field,28,31,49strongsolution,126,164simpleprocesses,91,215super(sub)martingale,50,183simulation,270swap,345space-homogeneous,58swaption,345,349spotrate,326–328spread,319Tanaka’sformula,222squarebracket,219,229Taylor’sformula,176squareintegrablemartingale,101,Termstructure,325184,230,231transformation,32stationarydistribution,170,355,385transforms,37StieltjesIntegral,10transient,81stochasticdifferential,108transitionfunction,135stochasticdifferentialequations,123transitionprobabilities,67stochasticexponential,128,131,224,uniformintegrability,185225,236,256usualconditions,50stochasticintegral,232stochasticintegraldiscretetime,84variation,4,6,8stochasticintensity,255,368variationofmartingales,200stochasticlogarithm,128,130,236,Vasicek’sModel,328248stochasticprocesses,23,47weaksolution,136,137,207stochasticsets,241WhiteNoise,124stochasticvolatility,307stoppingtime,25,51,68,242Yamada-Watanabe,134,174straddle,319yieldcurve,323,330StratanovichCalculus,145