Introduction to the theory of random.pdf

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1、IntroductiontothetheoryofrandomprocessesN.V.KrylovAuthoraddress:SchoolofMathematics,UniversityofMinnesota,Minneapo-lis,MN,55455E-mailaddress:krylov@math.umn.edu2000MathematicsSubjectClassification.Primary60-01;Secondary60G99.TheauthorwassupportedinpartbyNSFGrantDMS–9876586.Abstract.These

2、lecturenotesconcentrateonsomegeneralfactsandideasofthetheoryofstochasticprocesses.ThemainobjectsofstudyaretheWienerprocess,thestationaryprocesses,theinfinitelydivisibleprocesses,andtheItˆostochasticequations.Althoughitisnotpossibletocoverevenanoticeableportionofthetopicslistedaboveinasho

3、rtcourse,theauthorsincerelyhopesthatafterhavingfollowedthematerialpresentedherethereaderacquiresagoodunderstandingofwhatkindofresultsareavailableandwhatkindoftechniquesareusedtoobtainthem.Thesenotesareintendedforgraduatestudentsandscientistsinmathematics,physicsandengineeringinterestedi

4、nthetheoryofRan-domProcessesanditsapplications.ContentsPrefacexiChapter1.Generalities1§1.Someselectedtopicsfromprobabilitytheory1§2.SomefactsfrommeasuretheoryonPolishspaces5§3.Thenotionofrandomprocess14§4.Continuousrandomprocesses16§5.Hintstoexercises25Chapter2.TheWienerProcess27§1.Brow

5、nianmotionandtheWienerprocess27§2.SomepropertiesoftheWienerprocess32§3.Integrationagainstrandomorthogonalmeasures39§4.TheWienerprocesson[0,∞)50§5.MarkovandstrongMarkovpropertiesoftheWienerprocess52§6.ExamplesofapplyingthestrongMarkovproperty57§7.Itˆostochasticintegral61§8.Thestructureof

6、Itˆointegrablefunctions65§9.Hintstoexercises69Chapter3.Martingales71viiviiiContents§1.Conditionalexpectations71§2.Discretetimemartingales78§3.Propertiesofmartingales81§4.Limittheoremsformartingales87§5.Hintstoexercises92Chapter4.StationaryProcesses95§1.Simplestpropertiesofsecond-orderst

7、ationaryprocesses95§2.Spectraldecompositionoftrajectories101§3.Ornstein-Uhlenbeckprocess105§4.Gaussianstationaryprocesseswithrationalspectraldensities112§5.RemarksaboutpredictingGaussianstationaryprocesseswithrationalspectraldensities117§6.StationaryprocessesandtheBirkhoff-Khinc