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首页 Stability of a random diffusion with nonlinear d…

Stability of a random diffusion with nonlinear dri

Stability of a random diffusion…

jeff
2010-06-07 0人阅读 举报 0 0 暂无简介

简介:本文档为《Stability of a random diffusion with nonlinear dripdf》,可适用于工程科技领域

StatisticsProbabilityLetters()–Stabilityofarandomdi�usionwithnonlineardriftFubaoXiDepartmentofAppliedMathematics,BeijingInstituteofTechnology,Beijing,People’sRepublicofChinaReceivedJunereceivedinrevisedformDecemberAbstractForthesolutiontoarathergeneralnonlinearstochasticdi�erentialequationwithMarkovianswitching,werstproveitsFellercontinuityandtheexistenceanduniquenessofinvariantmeasurebythecouplingmethod,thendiscussitsstabilityintotalvariationnormbytheFoster–Lyapunovinequalityc©ElsevierBVAllrightsreservedMSC:JKeywords:CouplingFellercontinuityStabilityTotalvariationFoster–LyapunovinequalityIntroductionLet(X(t)Z(t))beastrongMarkovprocesswiththephasespaceRd×N,whereN:={:::n}TherstcomponentX(t)satisesthefollowinggeneralnonlinearstochasticdi�erentialequationwithMarkovianswitchingdX(t)=b(X(t)Z(t))dt�(X(t)Z(t))dB(t):()Let(FP)beaprobabilityspace,and{Ft}anincreasingfamilyofsub�algebrasofF,andB(t)anFtadaptedRdvaluedBrownianmotionandalsoamartingalewithrespectto{Ft}ThesecondcomponentZ(t)whichisindependentofB(t)andFtadapted,isarightcontinuousMarkovchainwithnitestatespaceNandQmatrix(qkl)(seeChen,)Moreover,supposethatqkl¿forallk�=lSincethereexiststherandomcomponentZ(t)inthedriftanddi�usioncoe>cientsof(),thesolutionX(t)to()canbecalledarandomdi�usionForx∈Rdand�=(�ij)∈SdEmailaddress:xifubaoxinhuanetcom(FXi)$seefrontmatterc©ElsevierBVAllrightsreserveddoi:jsplFXiStatisticsProbabilityLetters()–(thespaceofd×dmatrices),dene|x|=(d∑i=|xi|)=|�|=d∑ij=|�ij|=:Fortheexistenceanduniquenessofthesolutionto()wemakethefollowingrathergeneralassumptionsAssumptionBothb(xk)and�(xk)satisfytheLipschitzconditionandthelineargrowthconditionlocallyasfollows:ForanyM¿,thereisanHM¿suchthat|b(xk)−b(yk)||�(xk)−�(yk)|HM|x−y|()and|b(xk)||�(xk)|HM(|x|)()forallxy∈U(M)andk∈N,whereU(M):={x∈Rd:|x|M}IfAisavectorormatrix,weuseA∗todenoteitstransposeFor(xk)∈Rd×N,seta(xk)=�(xk)�(xk)∗LetC(Rd×R×NR)denotethefamilyofallnonnegativefunctionsV(xtk)onRd×R×Nwhicharetwicedi�erentiableinxandoncedi�erentiableint,whereR=∞)AsinMao(),wedeneanoperatorLonC(Rd×R×NR)asfollows:LV(xtk)=tV(xtk)d∑ij=aij(xk)xixjV(xtk)d∑i=bi(xk)xiV(xtk)∑l∈Nqkl(V(xtl)−V(xtk)):()AssumptionThereexistsanonnegativefunctionV(xk)onRd×Nwhichistwicecontinuouslydi�erentiableinxsuchthatforsomeconstant#¿,LV(xk)#V(xk)for(xk)∈Rd×N()inf{V(xk):|x|¿M}→∞asM→∞:()RemarkWecanprovethat()inAssumptionisequivalenttothatforsomeconstants#¿and$¿,LV(xk)#V(xk)$for(xk)∈Rd×N:()Indeed,()obviouslyimplies()Conversely,if()issatised,then()issatisedforfunctionV(xk)$=#UnderAssumptionsand,adoptingtheproofmethodof(TheoreminChapterIIIofKhas’minskii())or(TheoremofMeynandTweedie()),andsimultaneouslyusingLemmainChapterIIofSkorohod(),wecanprovethatstochasticdi�erentialequation(SDE)FXiStatisticsProbabilityLetters()–()hasauniquecontinuouslyregular(ienonexplosive)solutionX(t)InthesequelwewillalwaysassumethatAssumptionsandholdHence()andZ(t)togetherdetermineauniquestrongMarkovprocess(X(t)Z(t))Forthesakeofconvenience,weshallalsosaythat(X(t)Z(t))isthesolutiontoEq()Stabilityforstochasticdi�erentialequationswithMarkovianswitchinghasbeenwellstudiedbymanyauthors(seeBasaketal,JiandChizeck,Mao,andthereferencestherein)RelatedworkonstabilityofdiscretetimeARprocesseswithMarkovianswitchinghasbeendonebyYaoandAttali()Basaketal()studiedthestabilityindistributionforthemodel()withalineardrift,thatisb(xk)=A(k)xforevery(xk)∈Rd×N,whereA(k)∈Sdforeachk∈NTheyprovedthatthetransitionprobabilityP(t(xk)·)of(X(t)Z(t))convergesweaklytosomeprobabilitymeasure(·)ast→∞,forevery(xk)∈Rd×NPreviously,BasakandBhattacharya()discussedthesimilarproblemsinthedi�usioncontext,thatis,theparticularcaseof()withZ(t)removedInthepresentpaper,weconsidertheEq()witharathergeneralnonlineardrift,andshowhowtoconstructitscoupledsolutionsOnthebasisofthecouplingresults,weprovetheFellercontinuityof(X(t)Z(t))andtheexistenceanduniquenessofinvariantmeasurefor(X(t)Z(t))Finally,byvirtueoftheFoster–Lyapunovinequalityandthesuccessofcoupling,wediscussthestabilityintotalvariationnormfor(X(t)Z(t))(moreprecisely,theconvergenceinthetotalvariationnormofitstransitionprobabilityP(t(xk)·)toitsuniqueinvariantmeasure(·)ast→∞,forevery(xk)∈Rd×N)Sincetheconvergenceintotalvariationnormleadstotheweakconvergence,sothestabilityintotalvariationnormimpliesthestabilityindistributionCouplingandFellercontinuityInthissectionweconstructthecouplingof(X(t)Z(t))anditselfForasubsequentuse,wealsoprovetheFellercontinuityof(X(t)Z(t))bymakinguseofthecouplingmethodandatruncationargumentNowwerstconstructthebasiccouplingofthediscretecomponentZ(t)anditselfasfollowsLet(Z(t)Z′(t))betheMarkovprocesswithphasespaceN×Nandcouplingoperator(seeChen,fordetails)f(kl)=∑m(qkm−qlm)(f(ml)−f(kl))∑m(qlm−qkm)(f(km)−f(kl))∑mqkm∧qlm(f(mm)−f(kl))wherefisaboundedfunctiononN×NSetS=inf{t¿:Z(t)=Z′(t)}thenZ(t)andZ′(t)willmovetogetherfromSonwardFromChen()weknowthatonecanconstructmanykindsofcouplingsofZ(t)anditselfHowever,inthesequelweshallalwayslet(Z(t)Z′(t))bejustthebasiccouplingconstructedaboveNotingthatZ(t)isanitestateMarkovFXiStatisticsProbabilityLetters()–chainwithQmatrix(qkl)satisfyingthatqkl¿forallk�=l,andusingCorollaryofChen(),itisnotdi>culttoprovethefollowinglemmaLemmaForanykl∈N,wehaveP(kl)(S¡∞)=whereP(kl)denotesthedistributionof(Z(t)Z′(t))startingfrom(kl)AssumptionForeachk∈N,assumethatc(xyk)isad×dmatrixsuchthata(xyk)=(a(xk)c(xyk)c(xyk)∗a(yk))isnonnegativedeniteforany(xyk)∈Rd×Rd×NForafamilyof{c(xyk):k∈N}satisfyingAssumption,setA(xyk)=a(xk)a(yk)−c(xyk)OA(xyk)=〈x−yA(xyk)(x−y)〉=|x−y|x�=yB̂(xyk)=〈x−yb(xk)−b(yk)〉forxy∈Rdandk∈N,where〈··〉denotestheinnerproductinRdForeachfamilyof{c(xyk):k∈N}satisfyingAssumption,wecanconstructacouplingof(X(t)Z(t))anditselfasfollowsRecallthat(Z(t)Z′(t))isthebasiccouplingconstructedaboveSeta(txZ(t)yZ′(t))=(a(xZ(t))c(txZ(t)yZ′(t))c(txZ(t)yZ′(t))∗a(yZ′(t)))wherec(txZ(t)yZ′(t))=IS∞)(t)c(xyZ(t))Thenlet(X(t)Y(t))satisfySDEinRdd(X(t)Y(t))=�(tX(t)Z(t)Y(t)Z′(t))dW(t)(b(X(t)Z(t))b(Y(t)Z′(t)))dt()where�(txZ(t)yZ′(t))�(txZ(t)yZ′(t))∗=a(txZ(t)yZ′(t))andW(t)isaBrownianmotionindependentof(Z(t)Z′(t))inRdLetP(xkyl)denotethedistributionofthecoupling(X(t)Z(t)Y(t)Z′(t))startingfrom(xkyl),E(xkyl)denotethecorrespondingexpectationSetT=inf{t¿:X(t)=Y(t)Z(t)=Z′(t)}thenTisthecouplingtimeof(X(t)Z(t))and(Y(t)Z′(t))FXiStatisticsProbabilityLetters()–NowwedeneametriconRd×Nasfollows:((xm)(yn))=((xy)d(mn))=where(xy)=|x−y|=(d∑i=(xi−yi))=d(mn)={m=nm�=n:LetB(Rd×N)bethe�algebraonRd×Ngeneratedby(··)WedenotethetransitionprobabilitiesofthestrongMarkovprocess(X(t)Z(t))by{P(t(xk)A):t¿(xk)∈Rd×NA∈B(Rd×N)}De�nitionForp¿,wedenetheWassersteinmetricbetweenP(t(xk)·)andP(t(yk)·)asfollows:Wp(P(t(xk)·)P(t(yk)·))=infQ∫p((xm)(yn))Q(dxdmdydn)=pwhereQvariesoverallcouplingprobabilitymeasureswithmarginalsP(t(xk)·)andP(t(yk)·)TheoremThefamilyoftransitionprobabilities{P(t(xk)·):(xk)∈Rd×N}ofthestrongMarkovprocess(X(t)Z(t))isFellercontinuousBeforeproceedingtotheproofofTheorem,werstintroducesomenotationandprovetwousefullemmasForanM¿,setM=inf{t¿:(X(t)Z(t))�∈U(M)×N}whereU(M)wasdenedinAssumptionLetP(xk)denotethedistributionof(X(t)Z(t))startingfrom(xk),E(xk)denotethecorrespondingexpectationLemmaForanygivenM¿,thefamilyoftransitionprobabilities{PM(t(xk)·):(xk)∈U(M)×N}ofthetruncatedprocess(X(t∧M)Z(t∧M))isFellercontinuousProofSinceNhasthediscretemetric,weneedonlytoprovethat,foranyt¿,xy∈U(M)andk∈N,PM(t(xk)·)convergesweaklytoPM(t(yk)·)asx→yTothisend,bymeansofTheoremofChen(),itsu>cestoprovethat,W(PM(t(xk)·)PM(t(yk)·))→asx→y:()Inordertoprove(),weconstructasuitablecouplingof(X(t)Z(t))anditselfTodoso,accordingtotheconstructionmethodgivenabove,wenowonlyneedtochooseasuitablefamilyof{c(xyk):k∈N}satisfyingAssumptionTakec(xyk)=�(xk)�(xk)∗for(xk)∈Rd×N,thenthefamilyof{c(xyk):k∈N}satisesAssumptionTherefore,viaEq(),wegetaspeciccoupling(X(t)Z(t)Y(t)Z′(t))Similarlytotheproofof(TheoremofChenandLi()),setSm=inf{t¿:|Xt−Yt|¿m}TR=inf{t¿:|Xt||Yt|¿R}V=(TR∧Sm)∧M:FXiStatisticsProbabilityLetters()–UsingLemmainChapterIIofSkorohod(),wehaveE(xkyk)(Xt∧VYt∧V)=(xy)E(xkyk)∫t∧VtrA(Xu∧VYu∧VZu∧V)B̂(Xu∧VYu∧VZu∧V)du()forxy∈U(M)andk∈NOntheotherhand,bysomeelementarycomputationsand(),wehavetrA(xyk)=|�(xk)−�(yk)|HM|x−y|andB̂(xyk)HM|x−y|forxy∈U(M)andk∈NCombiningthesetwoestimateswith(),wegetE(xkyk)(Xt∧VYt∧V)(xy)(HMHM)∫tE(xkyk)(Xu∧VYu∧V)du:Thus,bytheGronwallinequality(cfLemmainChapterofFreidlinandWentzell()),weobtainE(xkyk)(Xt∧VYt∧V)(xy)exp(HM)t:Finally,lettingR↑∞,m↑∞,wearriveatE(xkyk)(X(t∧M)Y(t∧M))(xy)exp(HM)t:Clearly,thisisequivalenttoE(xkyk)((X(t∧M)Z(t∧M))(Y(t∧M)Z′(t∧M)))(xy)exp(HM)twhichimplies()TheproofisnishedLemmaForanyxedboundeddomainDinRdandt¿,wethenhaveP(xk)(Mt)→uniformlyover(xk)inD×NasM↑∞ProofNoting(),weknowthatfortheoperatorLdenedin(),L(V(xk)exp(−#t))=exp(−#t)LV(xk)−#V(xk)for(xk)∈Rd×NHence,byLemmainChapterIIofSkorohod(),wehaveE(xk)V(X(t∧M)Z(t∧M))exp−#(t∧M)−V(xk)=E(xk)∫t∧ML(V(X(s)Z(s))exp(−#s))dsFXiStatisticsProbabilityLetters()–andsoE(xk)V(X(t∧M)Z(t∧M))V(xk)exp(#t):Fromthis,wecaneasilyderivethatP(xk)(Mt)V(xk)exp(#t)(inf{V(xk):|x|¿M})−:Combiningthisestimatewith()inAssumption,weobtainthedesiredresultandproveLemmaProofofTheoremAccordingtothedenitionofFellercontinuity,itisenoughtoprovethatforanyboundedcontinuousfunctiong(xk)onRd×Nandt¿,E(xk)g(X(t)Z(t))isstillboundedcontinuousinbothxandkForanyM¿,bytheFellercontinuityofthetruncatedprocess(X(t∧M)Z(t∧M))provedinLemma,weknowthatE(xk)g(X(t∧M)Z(t∧M))isboundedcontinuousinbothxandkTherefore,itsu>cestoprovethatE(xk)g(X(t∧M)Z(t∧M))→E(xk)g(X(t)Z(t))()uniformlyover(xk)inD×NasM↑∞,whereDisanyxedboundeddomaininRdAsamatteroffact,bythedenitionofM,wehave|E(xk)g(X(t∧M)Z(t∧M))−E(xk)g(X(t)Z(t))|E(xk)|g(X(t∧M)Z(t∧M))−g(X(t)Z(t))|sup{|g(xk)|:(xk)∈Rd×N}P(xk)(Mt):CombiningthiswithLemmaandrecallingtheboundednessofg(xk),weknowthat()holdsThisterminatestheproofBeforeconcludingthissection,wepresentaresultonthesuccessfulcouplingwhichwillbeusedforthestudyofthestabilityintotalvariationnorminthenextsectionForaxedfamilyof{c(xyk):k∈N}satisfyingAssumption,choose(·)∈C(R)suchthat(r)¿sup{trA(xyk)−OA(xyk)B̂(xyk)=OA(xyk):(xy)=rk∈N}and#(·)∈C(R)suchthat#(r)inf{OA(xyk):(xy)=rk∈N}:DeneC(r)=exp∫r(u)uduf(r)=∫rC(s)−dsg(r)=∫rC(s)−ds∫sC(u)#(u)dur¿f(∞)=limr→∞f(r)g()=limr→g(r):FXiStatisticsProbabilityLetters()–TheoremIf#(r)¿on(∞),f(∞)=∞andg()¡∞,thenthecorrespondingcouplingissuccessful,thatisP(xkyl)(T¡∞)=(xk)�=(yl):ProofWerstprovethatP(xkyk)(T¡∞)=:()Todoso,followingChenandLi(),setSm=inf{t¿:|Xt−Yt|¿m}m¿Tn=inf{t¿:|Xt−Yt|¡n}n¿Tnm=Tn∧SmFnm()=−∫=nC(s)−ds∫msC(u)#(u)dunmnm¿:ThenwecanverifythatL˜Fnm((xy))¿forallthosexy∈Rdwith=n(xy)mandallk∈N,whereL˜isthecouplingdi�erentialoperatordenedbya(xyk)andb(xyk)=(b(xk)b(yk))∗asinChenandLi()Putr=(xy)UsingLemmainChapterIIofSkorohod(),wehaveE(xkyk)Fnm((Xt∧TnmYt∧Tnm))−Fnm(r)=E(xkyk)∫t∧TnmL˜Fnm((XuYu))du¿E(xkyk)(t∧Tnm):Then,usingtheproofofTheoremofChenandLi(),weobtain()Next,usingLemma,wehaveP(xkyl)(T¡∞)=P(xkyl)(T¡∞S¡∞)=P(xkyl)(S=TS¡∞)P(xkyl)(T¡∞|S=TS¡∞)P(xkyl)(S¡TS¡∞)P(xkyl)(T¡∞|S¡TS¡∞):Obviously,therstconditionalprobabilityequalsByvirtueofstrongMarkovpropertyand(),weknowthatthesecondconditionalprobabilityalsoequalsFinally,byLemmaagain,weproveTheoremFXiStatisticsProbabilityLetters()–StabilityintotalvariationInthissectionwewillinvestigatethefollowingstabilityintotalvariationnormfor(X(t)Z(t))De�nitionTheprocess(X(t)Z(t))issaidtobestableintotalvariationnormifthereexistsaprobabilitymeasure(·)suchthatitstransitionprobabilityP(t(xk)·)convergesto(·)intotalvariationnormast→∞forevery(xk)∈Rd×NRemarkIftheaboveconvergenceintotalvariationnormisreplacedbytheweakconvergence,thenonegetsthenotionofstabilityindistributionwhichwasstudiedinBasaketal()Sincetheconvergenceintotalvariationnormleadstotheweakconvergence,sothestabilityintotalvariationnormimpliesthestabilityindistributionMoreover,ifageneralMarkovprocessisstableintotalvariationnorm,itisalsooftensaidtobeergodic(seeSectionofMeynandTweedie())Soactuallywewillprovetheergodicityfortheprocess(X(t)Z(t))Inordertoformulateourstabilityresultprecisely,weshouldndthelimitingprobabilitymeasure(·)ForthiswerstusetheFellercontinuityobtainedintheprevioussectionandthefollowingFoster–Lyapunovdriftconditiontoprovethat(X(t)Z(t))hasaninvariantmeasurewhichjustisthelimitingprobabilitymeasure(·)WenowintroducetheFoster–Lyapunovdriftcondition:Forsome#,$¿,f(xk)¿,acompactsetC⊂Rd,andV(xk)¿whichistwicecontinuouslydi�erentiablefunctioninxonRd×N,LV(xk)−#f(xk)$IC×N(xk)(xk)∈Rd×N()whereListheoperatordenedin()andIC×NistheindicatorfunctionofthesetC×NTheoremSupposethat()holdsThenMarkovprocess(X(t)Z(t))hasaninvariantprobabilitymeasure(·),and∫f(xk)(dx×dk)$=#ProofFirst,weprovethat:forevery(xk)∈Rd×N,liminft→∞t∫tP(s(xk)C×N)ds¿#=$()limsupt→∞t∫tP(s)f(xk)ds$=#()whereP(s)f(xk)=∫P(s(xk)dy×dl)f(yl)RecallingthedenitionsofM,P(xk)andE(xk),andusingLemmainChapterIIofSkorohod()and(),wegetE(xk)V(X(t∧M)Z(t∧M))=V(xk)E(xk)∫t∧MLV(X(s)Z(s))dsV(xk)E(xk)∫t∧M(−#f(X(s)Z(s))$IC×N(X(s)Z(s)))ds:()Ontheonehand,itfollowsfrom()that#E(xk)(t∧M)V(xk)$∫tP(s(xk)C×N)ds:FXiStatisticsProbabilityLetters()–LettingM↑∞,weobtain#tV(xk)$∫tP(s(xk)C×N)dswhichimplies()Ontheotherhand,italsofollowsfrom()that#E(xk)∫t∧Mf(X(s)Z(s))dsV(xk)$t:LettingM↑∞,weobtain#∫tP(s)f(xk)dsV(xk)$twhichimplies()WenowcitethefollowingresultwhichwasprovedinFoguel()andStetter(),andstatedandusedintheproofofTheoremofMeynandTweedie()ForaFellerprocess(X(t)Z(t)),therearetwomutuallyexclusivepossibilities:eitheraninvariantprobabilitymeasure(·)exists,orlimt→∞sup:t∫t∫P(s(xk)C×B):(dx×dk)ds=()foranycompactsetC×B⊂Rd×N,wherethesupremumistakenoverallinitialdistributions:onRd×NBy(),weknowthat()isimpossibleCombiningthiswiththeFellercontinuityof(X(t)Z(t))provedinTheorem,weprovethat(X(t)Z(t))hasaninvariantprobabilitymeasure(·)UsingtheFatoulemmaand(),foranygiven¡M¡∞,wehave∫(f∧M)(xk)(dx×dk)=limsupt→∞∫(t∫tP(s)(f∧M)(xk)ds)(dx×dk)∫(limsupt→∞t∫tP(s)(f∧M)(xk)ds)(dx×dk)$=#:LettingM↑∞andusingthemonotoneconvergencetheorem,wegetthelastassertionThiscompletestheproofRemarkEq()canbeviewedastheresultofthefollowingnequations:dXk(t)=b(Xk(t)k)dt�(Xk(t)k)dB(t)k∈N()switchingfromonetoanotheraccordingtothemovementoftheMarkovchainZ(t)ThroughExamplebelow,wecanillustratethefollowinginterestingfactInordertoletthesolution(X(t)Z(t))ofEq()haveaninvariantprobabilitymeasure,itisnotnecessarytorequirethatforeachk∈N,thesolutionXk(t)ofthecorrespondingequationin()hasaninvariantprobabil

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Stability of a random diffusion with nonlinear dri

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