backward and forward equations for diffusion process

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1、BackwardandForwardequationsforDiffusionprocesses.ArkaP.GhoshDepartmentofStatisticsIowaStateUniversityAmes,IA50011-1210apghosh@iastate.edu(515)294-7851.February1,2010AbstractThissectionisdevotedtothediscussionoftwofundamental(partial)differentialequations,thatariseinthecontextofMarko

2、vdiffusionprocesses.Aftergivingabriefintroductionofcontinuous-timecontinuousstateMarkovprocesses,weintroducetheforwardandbackwardequation,andprovideaheuristicderivationoftheseequationsfordiffusionprocesses.Wealsodiscusssomeexamplesandfeaturesofthesetwoequations.Inthissectionwediscus

3、stwopartialdifferentialequations(PDE)thatariseinthetheoryofcontinuous-timecontinuous-stateMarkovprocesses,whichwasintroducedbyKolmogorovin1931.Here,wefocusonlyonMarkovdiffusionprocesses(seeSection2.1.6.1)anddescribetheforwardandbackwardequationforsuchprocesses.Theforwardequationisal

4、soknownasFokker-Planckequation(andwasalreadyknowninthephysicsliteraturebeforeKolmogorovformulatedthese).Webeginbyabriefintroductiontocontinuous-timecontinuous-stateMarkovprocesseswhicharecontinuousanalogsofDiscreteTimeMarkovChains(DTMC)andContinuousTimeMarkovChains(CTMC)discussede

5、arlierinSection2.1.1and2.1.2followedbysomebasicpropertiesofMarkovprocesses.Thenwestatethetwoequationsandprovidesketchesoftheproofs.Finally,weconcludethesectionwithsomespecificexamplesandfeaturesoftheseequations.Preliminaries.DiffusionprocesseshavebeendiscussedinSection2.1.6.1.Forsim

6、plicityoftheexposition,weconsiderthefollowingtime-homogeneousversionofthediffusionprocessforthissection:A(time-homogeneous)ltˆodiffusionisastochasticprocess{X(t)}satisfyingastochasticdifferentialequationoftheformdX(t)=b(X(t))dt+σ(X(t))dW(t),t>0;X(0)=x,(1)where{W(t)}isa(standard)Brown

7、ianmotionandb,σarefunctionsthatsatisfy:

8、σ(x)−σ(y)

9、

10、x−y

11、;x,y∈IR.1Itcanbeshownthatfor{FtW},{FtX}representingthefiltrationsgeneratedbyWandX,FX⊆FW.(2)ttMarkovproperty:hThediffusionsatisfiestheiMarkovproperty:Iffisaboundedmeasurablefunction,thenExf(X(t+h)

12、FtW=EX(t)[f(X(h)],wherethesuper

13、scriptintheexpectationrepresentst