Asymptotic Properties of Neutral Stochastic Differ

AsymptoticPropertiesofNeutralStochasticDierentialDelayEquationsXuerongMaoDepartmentofStatisticsandModellingScienceUniversityofStrathclydeGlasgowG11XH,Scotland,U.K.Abstract:Thispaperdiscussesasymptoticproperties,especiallyasymptoticstabilityofneutralstochasticdierentialdelayequations.Newtechniquesaredevelopedtocopewiththeneutraldelaycase,andtheresultsofthispaperaremoregeneralthantheauthor’searlierworkwithinthedelayequations.KeyWords:Lyapunovfunction,supermartingaleconvergencetheorem,It^o’sformula,asymp-toticstability.1.IntroductionStochasticdierentialequationshavebeenstudiedformorethanftyyearsandtherearemanybooksintheliteraturee.g.Arnold[1],Elworthy[2],Friedman[3],Has’minskii[6],Mao[10,11]andMohammed[15].Motivatedbythechemicalengineer-ingsystemsandthetheoryofaeroelasticity,Kolmanovskiietal.[7,8]introducedaclassofneutralstochasticfunctionaldierentialequations.Butsofarsuchequationshavebeendiscussedalittleunlikethedeterministicneutralequationsthathavebeenwellstudied(cf.Hale&Lunel[4],Hale&Meyer[5]).In1995Mao[12]initiatedthestudyofexponentialstabilityofaneutralstochasticfunctionaldierentialequationd[x(t)G(xt)]=f(xt;t)dt+g(xt;t)dB(t);(1:1)whileMao[13]employedtheRazumikhintechniquetoinvestigatetheexponentialstabilityofthisequation.TheresultsofMao[12,13]wereeditedbytheauthor’srecentbookMao[14]wheresomenewresultswerealsogiven.Inthispaperweconsideraneutralstochasticdierentialdelayequationd[x(t)G(x(t))]=f(x(t);x(t);t)dt+g(x(t);x(t);t)dB(t):(1:2)Clearlyequation(1.2)isaspecialcaseofequation(1.1)discussedinMao[12{14],butdelayequationsareoneofthemostimportantclassesoffunctionaldierentialequationsandneedspecialattention(cf.Hale&Lunel[4],Kolmanovskiietal.[7,8]).Theimportantfeaturesofthispaperare:Itisduetothisspecialcasethatweareabletodevelopnewtechniques,whicharecompletelydierentfromthoseusedinMao[12{14],toinvestigatetheasymptoticpropertiesofthesolutionsfortheneutraldelayequations.Thispaperconsidersnotonlythepth(p0)momentandalmostsureexponentialstabilitybutalsotheotherasymptoticpropertiese.g.asymptoticpolynomialPartiallysupportedbytheRoyalSocietyandtheEPSRC/BBSRC.1bounds,butMao[12{14]onlydiscussedthemeansquare(thecaseofp=2)andalmostsureexponentialstability.Duetothenewtechniquesdevelopedinthispaper,theresultsobtainedinthispaperareverygeneralanduseful.Thetheorydevelopedheregivesauniedtreat-mentforvariousasymptoticestimatese.g.exponentialandpolynomialbounds.ItisalsoworthtopointoutthatalthoughtheequationsdiscussedinMao[12],namelyequationsoftype(1.1),aremoregeneralthantheneutraldelayequation(1.2),theconditionsimposedtheremakethetheorynotapplicabletothedelayequation(pleaseseeSection5belowforthedetailedexplanation).Ontheotherhand,theresultsobtainedinMao[13,14]canbeappliedtothedelayequationbutarenotsoecientasthenewresultsinthispaper(pleaseagainseesection5belowfordetails).Inotherwords,bydevelopingnewtechniquesapplicabletothedelayequationweimproveourearlierresults.Fromtheseimportantfeaturesweseeclearlysignicantcontributionsofthispaper.2.KeyLemmasThroughoutthispaper,unlessotherwisespecied,welet(;F;fFtgt0;P)beacompleteprobabilityspacewithaltrationfFtgt0satisfyingtheusualconditions(i.e.itisrightcontinuousandF0containsallP-nullsets).LetB(t)=(B1(t);;Bm(t))Tbeanm-dimensionalBrownianmotiondenedontheprobabilityspace.LetjjdenotetheEuclideannorminRn.IfAisavectorormatrix,itstransposeisdenotedbyAT.IfAisamatrix,itstracenormisdenotedbyjAj=ptrace(ATA).Let0andC([;0];Rn)denotethefamilyofallcontinuousRn-valuedfunctionson[;0].LetCbF0([;0];Rn)bethefamilyofallF0-measurableboundedC([;0];Rn)-valuedrandomvariables=f():0g.Considerann-dimensionalneutralstochasticdierentialdelayequationd[x(t)G(x(t))]=f(x(t);x(t);t)dt+g(x(t);x(t);t)dB(t)(2:1)ont0withinitialdatafx():0g=2CbF0([;0];Rn).Heref:RnRnR+!Rn,g:RnRnR+!RnmandG:Rn!Rn.Weshallimposethefollowingstandinghypotheses:(H1)BothfandgsatisfythelocalLipschitzconditionandthelineargrowthcondition.Thatis,foreachk=1;2;,thereisack0suchthatjf(x;y;t)f(x;y;t)j_jg(x;y;t)g(x;y;t)jck(jxxj+jyyj)forallt0andthosex;y;x;y2Rnwithjxj_jyj_jxj_jyjk,andthereismoreoverac0suchthatjf(x;y;t)j_jg(x;y;t)jc(1+jxj+jyj)forall(x;y;t)2RnRnR+.(H2)Thereisaconstant2(0;1)suchthatjG(x)G(y)jjxyjforx;y2Rn2and,moreover,G(0)=0.Itisknown(cf.Mao[12])thatunderhypotheses(H1)and(H2),equation(2.1)hasauniquecontinuoussolutionont,whichisdenotedbyx(t;)inthispaper.Moreover,foreveryp0,Esupstjx(s;)jp1ont0:LetC2;1(RnR+;R+)denotethefamilyofallnonnegativefunctionsV(x;t)onRnR+whicharetwicecontinuouslydierentiableinxandonceint.ForeachV2C2;1(RnR+;R+),deneanoperatorLVfromRnRnR+toRbyLV(x;y;t)=Vt(xG(y);t)+Vx(xG(y);t)f(x;y;t)+12tracegT(x;y;t)Vxx(xG(y);t)g(x;y;t);whereVt(x;t)=@V(x;t)@t;Vx(x;t)=@V(x;t)@x1;;@V(x;t)@xn;Vxx(x;t)=@2V(x;t)@xi@xjnn:LetusstressthatLVisdenedonRnRnR+whileVonRnR+.WealsodenotebyL1(R+;R+)thefamilyofallfunctions:R+!R+suchthatR10(t)dt1.Furthermore,letC(Rn;R+)andC(RnR+;R+)denotethefamiliesofallcontinuousfunctionsfro