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arXiv:math/0203041v1[math.PR]5Mar2002LinearstochasticdifferentialequationswithfunctionalboundaryconditionsAureliAlabert∗DepartamentdeMatem`atiquesUniversitatAut`onomadeBarcelona08193Bellaterra,Cataloniae-mail:alabert@mat.uab.esMarcoFerrante†Dip.diMatematicaPuraedAppl.Universit`adegliStudidiPadovaviaBelzoni7,35131Padova,Italye-mail:ferrante@math.unipd.itFebruary1,2008AbstractWeconsiderlinearn-thorderstochasticdifferentialequationson[0,1],withlinearboundaryconditionssupportedbyafinitesubsetof[0,1].Westudysomefeaturesofthesolutiontotheseproblems,andespeciallyitsconditionalindependencepropertiesofMarkoviantype.AMSClassification:60H10,60J25∗SupportedbygrantsSGR99-87ofCIRITandBFM2000-0009ofDGESIC†SupportedbygrantCOFIN9901244421ofMURST1IntroductionItiswellknownthat,undersuitableLipschitzandgrowthconditionsonthecoefficients,aclassicalItˆostochasticdifferentialequationX(t)=ξ+Zt0b(s,X(s))ds+Zt0σ(s,X(s))dW(s),(1.1)whereWisaWienerprocessandξisaF0-measurablerandomvariableforagivennon-anticipatingfiltration{Ft,t≥0}ofW,hasauniquestrongsolutionwhichisaMarkovprocess.IfξisnotF0-measurableorthecoefficientsb,σarerandomandnon-adapted,thenanyreasonableinterpretationofXin(1.1)willnotbeanFt-adaptedprocessand,unlessσisaconstant,weneedtousesomeanticipatingstochasticintegraltogiveasensetotheequation.Inthesecases,thesolutionisnotaMarkovprocessingeneral.Stillanothersettingthatleadstoanticipationisthecaseofboundaryconditions.Thatmeans,thefirstvariableofthesolutionprocessisnolongeradatumoftheproblem,timerunsinaboundedinterval,sayfrom0to1,andweimposearelationh(X(0),X(1))=0betweenthefirstandthelastvariablesofthesolution.Inthissituation,thefactthatthesolutionwillnotbeMarkovianisquiteintuitive,sincethestrongrelationshipbetweenX(0)andX(1)willpreventtheindependenceofX(0)andX(1)fromholding,evenwhenconditioningtoX(a),a∈]0,1[,exceptmaybeinsomeveryparticularcases.Ontheotherhand,itmayalsoseemintuitivethatthefollowingweakerconditionalinde-pendencepropertycanholdtrue:Forany0≤ab≤1,theσ-fieldsσ{X(t),t∈[a,b]}andσ{X(t),t∈]a,b[c}areconditionallyindependentgivenσ{X(a),X(b)}.Wewilldenoteitbyσ{X(t),t∈[a,b]}σ{X(a),X(b)}σ{X(t),t∈]a,b[c}.(1.2)NowX(0)andX(1)areonthesamesideinrelation(1.2),sothattheboundaryconditiondoesnotseemtocausetheproblemseenabove.Butthefollowingexampleshowsthatthisiswrong:Example1.1Considertheproblem˙X(t)=f(X(t))dt+˙W(t),t∈[0,1]h(X(0),X(1))=0,wherethenoiseappearsadditively,andassumethatauniquesolutionexistsandthattheboundaryconditiongivenbyhdoesnotreducetoaninitialorfinalcondition.Then,relation(1.2)holdsifandonlyiff(x)=αx+β,forsomeconstantsαandβ.ThiswasprovedinNualartandPardoux[16].1Theprocessessatisfying(1.2)werecalledreciprocalprocessesbyS.Bernstein[5].TheconceptarosedirectlyfromE.Schr¨odingerideasontheformulationofquantummechanics.MorerecentresearchonsuchprocesseshasbeencarriedoutbyB.Jamison[13],A.Krener[15],R.Frezza,A.KrenerandB.Levy[10],M.Thieullen[20]andJ.C.Zambrini[21].Othernamescanbefoundintheliteraturetorefertothesameconcept.Areciprocalprocessisaone-parameterMarkovfieldinPaulL´evy’sterminology,andisalsocalledaquasi-Markovprocess,alocalMarkovprocessandaBernsteinprocess.WeshallsimplycallthemMarkovfields(seeDefinition4.1).Example1.2Considernowtheproblem¨X(t)+f(X(t),˙X(t))=˙W(t),t∈[0,1]X(0)=c1,X(1)=c2.Thisisasecondorderstochasticdifferentialequation,anditisnaturaltoaskforconditionalindependencepropertiesofthe2-dimensionalprocessY(t)=(˙X(t),X(t)),sinceX(t)hasC1paths,andthereforeitismeaninglesstolookforthiskindofpropertiesforX(t)itself.NualartandPardoux[17]provedthatifY(t)isaMarkovfield,then,asinExample1.1,fmustbeanaffinefunction.Moreover,iffisaffine,thenYisnotonlyaMarkovfield,butaMarkovprocess.Letuslookatthisexamplemoreclosely:Notethataboundaryconditionforasecondorderequationhasthegeneralformh(Y(0),Y(1))=(0,0),whereY(0)=(˙X(0),X(0))andY(1)=(˙X(1),X(1)).However,inExample1.2thetwoscalarconditionsdonotmixvaluesat0andvaluesat1ofY.Thesamehappens,forinstance,withtheNeumann-typeconditions˙X(0)=c1,˙X(1)=c2,andtheresultisthesame(YMarkovfield⇒faffine⇒YMarkovprocess).Fromtheseexamplesandotherequationsoffirstandsecondorderthathavebeenstudiedsofar(seee.g.[18],[2],[4],[3]),welearnthat1.TheMarkovianpropertiescanbeexpectedonlyin“linear”cases.2.ThespecificMarkovianpropertydependsontheactualformoftheboundarycondition.Itshouldalsobenotedthattherequirementoflinearityonthedriftcoefficientfisrelatedtothefactthatthenoiseappearsadditively.Shouldnotthisbethecase,theMarkovianpropertywouldoccurunderadifferentconditionwhichrelatesthedriftandthediffusioncoefficients(see[2]and[3]).Inthepresentpaperwewillconsiderlinearstochasticdifferentialequationsofarbitraryorderwithadditivewhitenoise.Ourboundaryconditionswillnotberestrictedtoinvolvethesolution2processattheendpointsofthetimeinterval,butwewillallowthemtoinvolvethevaluesatfinitelymanypointsinsidetheinterval.Theyareusuallycalledfunctionalorlateralboundaryconditions.Ourmaingoalistoseekwhichkindofconditionalindependencepropertiescanbeestablishedforthesolution.ApreliminaryworkinthisdirectionwaspublishedinAlabertandFerrante[1].Hereweconsiderablyrefinea
本文标题:Linear stochastic differential equations with func
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