$A simple theory for the study of SDEs driven by a fractional Brownian motion in dimension one$

A simple theory for the study of SDEs driven by a fractional Brownian motion in dimension one

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ar X iv :m at h/ 05 11 02 7v 4 [m ath .PR ] 18 O ct 20 07 A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one Ivan Nourdin Laboratoire de Probabilites et Modeles Aleatoires, University Pierre et Marie Curie Paris VI, Boıte courrier 188, 4 Place Jussieu, 75252 Paris Cedex 5, France Summary. We will focus – in dimension one – on the SDEs of the type dXt = ?(Xt)dBt + b(Xt)dt where B is a fractional Brownian motion. Our principal aim is to describe a simple theory – from our point of view – allowing to study this SDE, and this for any H ? (0, 1). We will consider several definitions of solutions and, for each of them, study conditions under which one has existence and/or uniqueness. Finally, we will examine whether or not the canonical scheme associated to our SDE converges, when the integral with respect to fBm is defined using the Russo-Vallois symmetric integral. Key words: Stochastic differential equation; fractional Brownian motion; Russo-Vallois integrals; Newton-Cotes functional; Approximation schemes; Doss-Sussmann transformation. MSC 2000: 60G18, 60H05, 60H20.

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AsimpletheoryforthestudyofSDEsdrivenbyafractionalBrownianmotion,indimensiononeIvanNourdinLaboratoiredeProbabilite´setMode`lesAle´atoires,UniversityPierreetMarieCurieParisVI,Boıˆtecourrier188,4PlaceJussieu,75252ParisCedex5,Francenourdin@ccr.jussieu.frSummary.Wewillfocus–indimensionone–ontheSDEsofthetypedXt=σ(Xt)dBt+b(Xt)dtwhereBisafractionalBrownianmotion.Ourprincipalaimistodescribeasimpletheory–fromourpointofview–allowingtostudythisSDE,andthisforanyH∈(0,1).Wewillconsiderseveraldeﬁnitionsofsolutionsand,foreachofthem,studyconditionsunderwhichonehasexistenceand/oruniqueness.Finally,wewillexaminewhetherornotthecanonicalschemeassociatedtoourSDEconverges,whentheintegralwithrespecttofBmisdeﬁnedusingtheRusso-Valloissymmetricintegral.Keywords:Stochasticdiﬀerentialequation;fractionalBrownianmotion;Russo-Valloisintegrals;Newton-Cotesfunctional;Approximationschemes;Doss-Sussmanntransformation.MSC2000:60G18,60H05,60H20.1IntroductionThefractionalBrownianmotion(fBm)B={Bt,t≥0}ofHurstindexH∈(0,1)isacenteredandcontinuousGaussianprocessverifyingB0=0a.s.andE[(Bt−Bs)2]=|t−s|2H(1)foralls,t≥0.ObservethatB1/2isnothingbutstandardBrownianmotion.Equality(1)impliesthatthetrajectoriesofBare(H−ε)-Ho¨ldercontinuous,foranyε>0smallenough.AsthefBmisselfsimilar(ofindexH)andhasstationaryincrements,itisusedasamodelinmanyﬁelds(forexample,inhydrology,economics,ﬁnancialmathematics,etc.).Inparticular,thestudyofstochasticdiﬀerentialequations(SDEs)drivenbyafBmisimportantinviewoftheapplications.But,beforeraisingthequestionofexistenceand/oruniquenessforthistypeofSDEs,theﬁrstdiﬃcultyistogiveameaningtotheintegralwithrespecttoafBm.Itisindeedwell-knownthatBisnotasemimartingalewhenH6=1/2.Thus,theItoˆorStratonovichcalculusdoesnotapplytothiscase.ThereareseveralwaysofbuildinganintegralwithrespecttothefBmandofobtainingachangeofvariablesformula.Letuspointoutsomeofthesecontributions:1.Regularizationordiscretizationtechniques.Since1993,RussoandVallois[31]havedevelopedaregularizationprocedure,whosephilosophyissimilartothediscretization.Theyintroduceforward(generalizingItoˆ),backward,symmetric(generalizingStratonovich,seeDeﬁnition3below)stochasticintegralsandageneralizedquadraticvariation.Theregularization,ordis-cretizationtechnique,forfBmandrelatedprocesseshavebeenperformedby[12,17,32,36],inthecaseofzeroquadraticvariation(correspondingtoH>1/2).NotealsothatYounginte-grals[35],whichareoftenusedinthiscase,coincidewiththeforwardintegral(butalsowiththebackwardorsymmetricones,sincecovariationbetweenintegrandandintegratorisalwayszero).Whentheintegratorhaspathswithﬁnitep-variationforp>2,forwardandbackward

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