Stability Properties of Constrained Jump-Diffusion

arXiv:math/0501014v1[math.PR]2Jan2005ElectronicJournalofProbabilityVol.7(2002)Paperno.12,pages1–36.JournalURL˜ejpecp/PaperURL˜ejpecp/EjpVol7/paper12.abs.htmlSTABILITYPROPERTIESOFCONSTRAINEDJUMP-DIFFUSIONPROCESSESRamiAtarDepartmentofElectricalEngineering,Technion,Haifa32000,IsraelAmarjitBudhirajaDepartmentofStatistics,UniversityofNorthCarolina,ChapelHill,NC27599-3260,USAAbstract:Weconsideraclassofjump-diffusionprocesses,constrainedtoapolyhedralconeG⊂IRn,wheretheconstraintvectorfieldisconstantoneachfaceoftheboundary.Theconstrainingmechanismcorrectsfor“attempts”oftheprocesstojumpoutsidethedomain.UnderLipschitzcontinuityoftheSkorohodmapΓ,itisknownthatthereisaconeCsuchthattheimageΓφofadeterministiclineartrajectoryφremainsboundedifandonlyif˙φ∈C.Denotingthegeneratorofacorrespondingunconstrainedjump-diffusionbyL,weshowthatakeyconditionfortheprocesstoadmitaninvariantprobabilitymeasureisthatforx∈G,Lid(x)belongstoacompactsubsetofCo.Keywords:Jumpdiffusionprocesses.TheSkorohodmap.Stabilitycone.Harrisrecurrence.AMSsubjectclassification:60J6060J75(34D20,60K25)SubmittedtoEJPonSeptember24,2001.FinalversionacceptedonMarch20,2002.ThisresearchwassupportedinpartbytheUS-IsraelBinationalScienceFoundationandthefundforthepromotionofresearchattheTechnion(RA),andtheIBMjuniorfacultydevelop-mentaward,UniversityofNorthCarolinaatChapelHill(AB).1IntroductionInthisworkweconsiderstabilitypropertiesofaclassofjump-diffusionprocessesthatareconstrainedtolieinaconvexclosedpolyhedralcone.LetGbeaconeinIRn,givenastheintersection∩iGiofthehalfspacesGi={x∈IRn:x·ni≥0},i=1,...,N,whereni,i=1,...,Naregivenunitvectors.ItisassumedthattheoriginisapropervertexofG,inthesensethatthereexistsaclosedhalfspaceG0withG∩G0={0}.Equivalently,thereexistsaunitvectora0suchthat{x∈G:x·a0≤1}(1.1)iscompact.Notethat,inparticular,N≥n.LetFi=∂G∩∂Gi.WitheachfaceFiweassociateaunitvectordi(suchthatdi·ni0).ThisvectordefinesthedirectionofconstraintassociatedwiththefaceFi.Theconstraintvectorfieldd(x)isdefinedforx∈∂Gasthesetofallunitvectorsintheconegeneratedby{di,i∈In(x)},whereIn(x).={i∈{1,...,N}:x·ni=0}.Underfurtherassumptionson(ni)and(di),onecandefineaSkorohodmapΓinthespaceofrightcontinuouspathswithleftlimits,inawaywhichisconsistentwiththeconstraintvectorfieldd.Namely,Γmapsapathψtoapathφ=ψ+ηtakingvaluesinG,sothatηisofboundedvariation,and,denotingthetotalvariationofηon[0,s]by|η|(s),dη(·)/d|η|(·)∈d(φ(·)).TheprecisedefinitionofΓandtheconditionsassumedaregiveninSection2.Theconstrainedjump-diffusionstudiedinthispaperisthesecondcomponentZofthepair(X,Z)ofprocessessatisfyingXt=z0+Zt0β(Zs)ds+Zt0a(Zs)dWs+Z[0,t]×Eh(δ(Zs−,z))[N(ds,dz)−q(ds,dz)]+Z[0,t]×Eh′(δ(Zs−,z))N(ds,dz),(1.2)Z=Γ(X).(1.3)Here,WandNarethedrivingm-dimensionalBrownianmotionandPoissonrandommeasureonIR+×E;β,aandδare(state-dependent)coefficientsandhisatruncationfunction(seeSection2fordefinitionsandassumptions).Forillustration,considerasaspecialcaseof(1.2),(1.3),thecasewhereXisaL´evyprocesswithpiecewiseconstantpathsandfinitelymanyjumpsoverfinitetimeintervals.ThenXt=x+Ps≤tΔXs,whereΔXs=Xs−Xs−.Inthiscase,ZisgivenasZt=x+Ps≤tΔZs,whereΔZscanbedefinedrecursivelyinastraightforwardway.Namely,ifZs−+ΔXs∈G,thenΔZs=ΔXs.Otherwise,Zs=Zs−+ΔXs+αd,whereα∈(0,∞),Zs∈∂G,andd∈d(Zs).Ingeneral,thissetofconditionsmaynothaveasolution(α,d),ormayhavemultiplesolutions.However,theassumptionsweputonthemap2Γwillensurethatthisrecursionisuniquelysolvable,andasaresult,thattheprocessZiswelldefined.Arelatedmodelforwhichrecurrenceandtransiencepropertieshavebeenstudiedexten-sivelyisthatofasemimartingalereflectingBrownianmotion(SRBM)inpolyhedralcones[3,8,11,12,13].Roughlyspeaking,aSRBMisaconstrainedversion,usinga“constrainingmechanism”asdescribedabove,ofaBrownianmotionwithadrift.Inarecentwork[1],sufficientconditionsforpositiverecurrenceofaconstraineddiffusionprocesswithastatede-pendentdriftand(uniformlynondegenerate)diffusioncoefficientswereobtained.UndertheassumptionofregularityofthemapΓ(asinCondition2.4below),itwasshownthatifthedriftvectorfieldtakesvaluesintheconeCgeneratedbythevectors−di,i=1,...,N,andstaysaway,uniformly,fromtheboundaryofthecone,thenthecorrespondingconstraineddiffusionprocessispositiverecurrentandadmitsauniqueinvariantmeasure.Thetechniqueusedtherecriticallyreliesoncertainestimatesontheexponentialmomentsoftheconstrainedprocess.ThecurrentworkaimsatshowingthatCplaystheroleofastabilityconeinamuchmoregeneralsettingofconstrainedjump-diffusionsforwhichonlythefirstmomentisassumedtobefinite.Thenaturaldefinitionofthedriftvectorfieldinthecaseofajump-diffusionis˜β.=Lid,whereLdenotesthegeneratorofarelated“unconstrained”jump-diffusion(see(2.6)),andiddenotestheidentitymappingonIRn.InthecaseofaL´evyprocesswithfinitemean,thedriftissimply˜β(x)=ExX1−x(whichisindependentofx).Ourbasicstabilityassumptionisthattherangeof˜βiscontainedin∪k∈INkC1,whereC1isacompactsubsetoftheinteriorofC.Underthisassumption,ourmainstabilityresultstates(Theorem2.13):ThereexistsacompactsetAsuchthatforanycompactC⊂G,supx∈CExτA∞,(1.4)whereτAisthefirsttimeZhitsA,andExdenotestheexpe