continuous stochastic processes

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1、Chapter15ContinuousStochasticProcessesItistheaimofthischaptertobrießyrecallbasicdeÞnitionsandresultsconcern-ingcontinuousstochasticprocesses.WealsodiscusstheItformula,theFeynman-KacformulaandexistenceanduniquenessofsolutionsofSDEsdrivenbyWienerprocesse

2、s.Thischapteristailoredtowardsthecontentpresentedinthisbook.Foramoredetaileddiscussion,alsocoveringSDEsdrivenbyPoissonprocessesandPoissonrandommeasures,thereaderisreferredtoe.g.Chap.1inPlatenandBruti-Liberati(2010).15.1StochasticProcesses15.1.1Stochast

3、icProcessIfnototherwisestated,throughoutthechapterweassumethatthereexistsacom-monunderlyingprobabilityspace(Ω,A,P)consistingofthesamplespaceΩ,thesigma-algebraorcollectionofeventsA,andtheprobabilitymeasureP,seeforinstanceShiryaev(1984).Onetypicallyobser

4、vesacollectionofrandomvariablesXt,Xt,...,whichdescribetheevolutionofÞnancialquantities,attheobservation01timest0

5、={1,2,...},orasubsetofit.HoweverinChaps.10and11,wealsoconsidermatrix-valueddiffusions.Definition15.1.1WecallafamilyX={Xt,t∈T}ofrandomvariablesXt∈dad-dimensionalstochasticprocess,wherethetotalityofitsÞnite-dimensionaldis-tributionfunctionsFXti1,...,Xtij

6、(xi1,...,xij)=P(Xti1≤xi1,...,Xtij≤xij)(15.1.1)forij∈{0,1,...},j∈N,xi∈dandti∈Tdeterminesitsprobabilitylaw.jjJ.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffusionswithApplications363toFinance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319-00747-2_15,

7、©SpringerInternationalPublishingSwitzerland201336415ContinuousStochasticProcessesWesetthetimesettotheintervalT=[0,∞)ifnototherwisestated.Onsomeoccasionsthetimesetmaybecometheboundedinterval[0,T]forT∈(0,∞)orasetofdiscretetimepoints{t0,t1,t2,...},wheret0

8、