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1、StochasticProcessesincontinuoustimeStefanGeissApril28,20092Contents1Introduction52Stochasticprocessesincontinuoustime72.1Somede?nitions..........................72.2Twobasicexamplesofstochasticprocesses...........122.3Gaussianprocesses........................142.4Brownianmotion.....................

2、....252.5Stoppingandoptionaltimes...................292.6AshortexcursiontoMarkovprocesses.............333Stochasticintegration353.1De?nitionofthestochasticintegral...............363.2It?o’sformula............................533.3ProofofIto?’sformulainasimplecase.............634Stochasticdi?erential

3、equations674.1Whatisastochasticdi?erentialequation?...........674.2StrongUniquenessofSDE’s...................704.3ExistenceofstrongsolutionsofSDE’s..............734.4SolutionsofSDE’sbyatransformationofdrift.........754.5Weaksolutions..........................8134CONTENTSChapter1IntroductionOnegoalo

4、fthelectureistostudystochasticdi?erentialequations(SDE’s).Soletusstartwitha(hopefully)motivatingexample:AssumethatXtisthesharepriceofacompanyattimet≥0whereweassumewithoutlossofgeneralitythatX0:=1.TogetanideaofthedynamicsofXletusconsidertherelativeincrements(thesearetheincrementswhicharerelevantin?n

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-wisetheproblemgetstoodi?cult.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