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1、IEORE4707:FinancialEngineering:Continuous-TimeModelsFall2010°c2010byMartinHaughIntroductiontoStochasticCalculusThesenotesprovideanintroductiontostochasticcalculus,thebranchofmathematicsthatismostidenti¯edwith¯nancialengineeringandmathematical¯nance.Wewilligno
2、remostofthetechnical"detailsandtakeanengineering"approachtothesubject.Wewillcovermorematerialthanisstrictlynecessaryforthiscourse.Anymaterialthatisnotrequired,however,shouldbeofvalueforothercoursessuchasTermStructureModels.Wemakethefollowingassumptionsthrou
3、ghout.²Thereisaprobabilitytriple(•;F;P)where{Pisthetrue"orphysicalprobabilitymeasure{•istheuniverseofpossibleoutcomes.Weuse!2•torepresentagenericoutcome,typicallyasamplepath(s)ofastochasticprocess(es).{theset1Frepresentsthesetofpossibleeventswhereaneventisas
4、ubsetof•.²Thereisalsoa¯ltration,fFtgt¸0,thatmodelstheevolutionofinformationthroughtime.Soforexample,ifitisknownbytimetwhetherornotanevent,E,hasoccurred,thenwehaveE2Ft.Ifweareworkingwitha¯nitehorizon,[0;T],thenwecantakeF=FT.²Wealsosaythatastochasticprocess,Xt,
5、isFt-adaptedifthevalueofXtisknownattimetwhentheinformationrepresentedbyFtisknown.AlltheprocessesweconsiderwillbeFt-adaptedsowewillnotbothertostatethisinthesequel.²Inthecontinuous-timemodelsthatwewillstudy,itwillbeunderstoodthatthe¯ltrationfFtgt¸0willbethe¯ltr
6、ationgeneratedbythestochasticprocesses(usuallyaBrownianmotion,Wt)thatarespeci¯edinthemodeldescription.²Sincethesenotesregularlyrefertonumerairesandequivalentmartingalemeasures(EMMs),readersshouldbefamiliarwiththeseconceptsinadvance.MyMartingalePricingTheoryle
7、cturenote,forexample,coversthesetopicsinadiscrete-time,discrete-spaceframework.InSection12ofthesenoteswewilldiscussmartingalepricingtheoryinthecontinuous-timesettingandstatewithoutproof2thetwoFundamentalTheoremsofAssetpricing.1MartingalesandBrownianMotionDe¯n
8、ition1Astochasticprocess,fWt:0·t·1g,isastandardBrownianmotionif1.W0=02.Ithascontinuoussamplepaths3.Ithasindependent,normally-distributedincrements.1Technically,Fisa¾-algebra.2TheseTheorem
