time homogeneous scalar diffusions

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1、Chapter16Time-HomogeneousScalarDiffusionsInthisbook,wepursuemostlyaprobabilisticapproach,essentiallyoriginatedinIt(1944),whosetouttogeneratediffusionsdirectlyfromgivenBrownianmotions.Thischapterrefersmoretoananalyticapproachtodiffusions,tracingitsoriginbacktoKolmogorov(1932)andFeller(1936).The

2、approachallowsustoobtaintransitiondensitiesbysolvingKolmogorovequations,seeSect.15.8.FollowingHulley(2009),weexplainbelowhowtoobtainsomeexplicitsolutionsforfunctionalsofscalardiffusions,especiallyfunctionalsassociatedwithstoppingtimes.16.1BasicDefinitionsWiththefollowingdeÞnitionwefollowHulley(

3、2009).Definition16.1.1FixanintervalI⊆,withleftend-pointl≥−∞andrightend-pointr≤∞.DenotethespaceofcontinuousI-valuedpathsbyΩ:=C(+,I)andletXbethecoordinatemappingprocessonthisspace,deÞnedbyXt(ω):=ω(t),+000forallω∈Ωandt∈.DeÞnetheÞltrationA=(At)t∈+,bysettingAt:=σ{Xs

4、s≤t},forallt∈+,aswellastheσ-

5、algebraA0∞:=σ{Xt

6、t∈+}.Theshiftoperatorsϑ=(ϑt)t∈+areconstructed,bysetting(ϑtω)(s):=ω(t+s),forallω∈Ωandt,s∈+.Finally,letP={Px

7、x∈I}beafamilyofprobabilitymeasureson(Ω,A0∞),satisfying:(i)x→Px(A)ismeasurable,forallA∈A0∞;(ii)Px(X0=x)=1,forallx∈I;(iii)Ex(η◦ϑσ

8、A0+)=EX(η)Px-a.s.,σσforallboundedA0∞-m

9、easurablerandomvariablesη,andallA0-stoppingtimesσ.Thetuple(Ω,A0,A0,X,ϑ,P)isthencalledacanonicaldiffusiononI.∞TheÞltrationAusedinDeÞnition16.1.1isnotnecessarilyright-continuousorcomplete.Weremedythisbyintroducingtheright-continuousÞltrationJ.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffus

10、ionswithApplications389toFinance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319-00747-2_16,©SpringerInternationalPublishingSwitzerland201339016Time-HomogeneousScalarDiffusions@+++0+A=(At)t∈+,deÞnedbysettingAt:=>0At+,forallt∈.Next,thefamilyofnull-setsisintroduced:A:=N⊆Ω

11、N⊆A,forsomeA∈A0satisfy

12、ingP∞x(A)=0,∀x∈I.++TheÞltrationA=(At)t∈+isconstructedbysettingAt:=At∨N,forallt∈.SincenoneoftheaboveaffectsthestrongMarkovpropertyofX,asexpressedbyDeÞnition16.1.1(iii),weshallhenceforthregard(Ω,A∞,A,X,ϑ,P)asthediffusionunderconsiderati