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1、StochasticProcessesincontinuoustimeStefanGeissApril28,20092Contents1Introduction52Stochasticprocessesincontinuoustime72.1Somedefinitions..........................72.2Twobasicexamplesofstochasticprocesses...........122.3Gaussianprocesses........................142.4Brownianmotion.....................
2、....252.5Stoppingandoptionaltimes...................292.6AshortexcursiontoMarkovprocesses.............333Stochasticintegration353.1Definitionofthestochasticintegral...............363.2Itˆo’sformula............................533.3ProofofItoˆ’sformulainasimplecase.............634Stochasticdifferential
3、equations674.1Whatisastochasticdifferentialequation?...........674.2StrongUniquenessofSDE’s...................704.3ExistenceofstrongsolutionsofSDE’s..............734.4SolutionsofSDE’sbyatransformationofdrift.........754.5Weaksolutions..........................8134CONTENTSChapter1IntroductionOnegoalo
4、fthelectureistostudystochasticdifferentialequations(SDE’s).Soletusstartwitha(hopefully)motivatingexample:AssumethatXtisthesharepriceofacompanyattimet≥0whereweassumewithoutlossofgeneralitythatX0:=1.TogetanideaofthedynamicsofXletusconsidertherelativeincrements(thesearetheincrementswhicharerelevantinfin
5、ancialmarkets)Xt+∆−Xt∼b∆+σYt,∆Xtwithb∈IR,σ>0,and∆>0beingsmall.Hereb∆describesageneraltrendandσYt,∆somerandomevents(perturbations).Askingseveralpeopleaboutthisapproachweprobablygetanswerslikethat:•Statisticians:therandomvariablesYt,∆shouldbecenteredGaussianrandomvariables.•Mathematicians:theperturba
6、tionsshouldnothaveamemory,other-wisetheproblemgetstoodifficult.HenceYt,∆isindependentfromXt.•Then,inaddition,probablybothofthemagreetoassumethattheperturbationsbehaveadditively,thatmeansYt,∆=Yt,∆+Yt+∆,∆222sothatvar(Yt,∆)=∆isagoodchoice.AnapproachlikethisyieldstothefamousBlack-Scholesoptionpricingmode
7、l.Isitpossibletomakeoutofthisacorrectmathematicaltheory?Yesitis,ifweproceedforexampleinthefollowingway:Step1:TherandomvariablesYt,∆willbereplacedbyacontinuoustimestochasticprocessW=(Wt)t≥0,calledBrownianmot
