多维带跳的随机泛函微分方程的保序性黄兴1，王凤雨1,21北京师范大学数学科学学院，北京1008752斯旺西大学数学学院，斯旺西SA28PP摘要：本文给出了非Lipschitzian系数条件下由布朗运动和泊松过程驱动的多维随机泛函微分方程保序性的充要条件.充分条件推广了一维和多维不带延迟的随机微分方程的现有结果,而必要性即使在上述特殊情形下也是最新成果.关键词：概率论,保序性,比较定理,随机泛函微分方程中图分类号：60J75,47G20,60G52Order-PreservationforMultidimensionalStochasticFunctionalDifferentialEquationswithJumpHUANGXing1,WANGFengYu1212DepartmentofofMathematicalSciences,BeijingNormalUniversity,Beijing100875DepartmentofMathematics,SwanseaUniversity,SwanseaSingletonPark,SA28PPAbstract:Sufficientandnecessaryconditionsarepresentedfortheorder-preservationofmultidimensionalstochasticfunctionaldifferentialequationswithnon-LipschitziancoefficientsdrivenbytheBrownianmotionandPoissonprocesses.Thesufficiencyoftheconditionsextendsandimprovessomeknowncomparisontheoremsderivedrecentlyforone-dimesionalequationsandmultidimensionalequationswithoutdelay,andthenecessityisneweveninthesespecialsituations.Keywords:Theoryofprobability,order-preservation,comparisontheorem,stochasticfunctionaldifferentialequation.0IntroductionTheorder-preservationofstochasticprocessesiscrucialsinceitenablesonetocontrolcomplicatedprocessesbyusingsimplerones.Foralargeclassofdiffusion-jumptypeMarkovprocessesonRd,theorder-preservationpropertyhasbeenwelldescribedinthedistribution基金项目：Lab.Math.Com.Sys.,NNSFC(11131003,11126350),SRFDP,andtheFundamentalResearchFundsfortheCentralUniversities.作者简介：黄兴（1988-），男，硕士，主要研究方向：概率论，随机分析。通信作者：王凤雨（1966-），男，教授，主要研究方向：概率论，微分几何，统计物理，泛函分析等。-1-¯b¯¯sense(see[1],[2]andreferencestherein),seealso[3]forastudyofsuperprocesses.Toderivethepathwiseorder-preservation,oneestablishesthecomparisontheoremforstochasticdifferentialequations(SDEs),whichgoesbackto[4],[5].Thestudyofcomparisontheoremforone-dimensionalSDEsisnowverycomplete,seee.g.[6],[7],[8],[9],[10],[11],[12],[13],[14]andreferenceswithin.EquationsconsideredinthesereferencesincludeforwardorbackwardSDEswithjumpandwithdelay.Theaimofthisnoteistoprovideasharpcriteriononthecomparisontheoremformultidimensionalstochasticfunctionalstochasticdifferentialequations(SFDEs),whichisyetunknownintheliterature.Throughoutthepaper,wefixaconstantr0≥0andanaturalnumberd≥1.LetC=ξ=(ξ1,···,ξd):[−r0,0]→Rdiscadlag.Recallthatapathiscalledcadlag,ifitisright-continuoushavingfiniteleftlimits.Foranyξ∈C,wehavedξ∞=sup|ξi(s)|<∞.i=1s∈[−r0,0]Thenundertheuniformnorm·∞thespaceCiscompletebutnotseparable.TomakeCaPolishspace,wetaketheSkorohodmetricratherthantheuniformmetric.Foranycadlagf:[−r0,∞)→Rdandt≥0,weletft∈Cbesuchthatft(θ)=f(θ+t)forθ∈[−r0,0],anddefineft−∈Cfort>0suchthatft−(θ)=f((t+θ)−):=lims↑t+θf(s)forθ∈[−r0,0].Wecall(ft)t≥0thesegmentof(f(t))t≥−r0.Now,letB(t)beanm-dimensionalBrownainmotion,andletN(ds,dz)beaPoissoncountingprocesswithcharacteristicmeasureνonameasurablespace(E,E),withrespecttoacompletefilteredprobabilityspace(Ω,F,{Ft}t≥0,P).WeassumethatBandNareindependent.Wewillconsidertheorder-preservationofSFDEsdrivenbyBandN.Tocharacterizethenon-LipshitzregularityofcoefficientsintheSDDEs,weintroducethefollowingclassofcontrolfunctions:U=u∈C1((0,∞);[1,∞)):01dssu(s)=∞,limsu(s)2=0,s↓...