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

Ivan Nourdin

Description

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.

• let ? ?


• then


• has been


• newton-cotes functional


• functional ∫


• russo-vallois symmetric

Sujets

• Nur al-Din
• Has Been

Informations

• Publié par: pefav
• Nombre de lectures: 12
• Langue: English

Extrait

AsimpletheoryforthestudyofSDEsdrivenbya fractionalBrownianmotion,indimensionone IvanNourdin LaboratoiredeProbabilite´setMode`lesAle´atoires,UniversityPierreetMarieCurieParisVI, Boıˆtecourrier188,4PlaceJussieu,75252ParisCedex5,France [e-mail] Summary. Wewillfocus–indimensionone–ontheSDEsofthetype dX t = σ ( X t ) dB t + b ( X t ) dt where B isafractionalBrownianmotion.Ourprincipalaimistodescribeasimpletheory–fromourpoint ofview–allowingtostudythisSDE,andthisforany H ∈ (0 , 1).Wewillconsiderseveraldefinitions ofsolutionsand,foreachofthem,studyconditionsunderwhichonehasexistenceand/oruniqueness. Finally,wewillexaminewhetherornotthecanonicalschemeassociatedtoourSDEconverges,whenthe integralwithrespecttofBmisdefinedusingtheRusso-Valloissymmetricintegral. Keywords: Stochasticdifferentialequation;fractionalBrownianmotion;Russo-Valloisintegrals; Newton-Cotesfunctional;Approximationschemes;Doss-Sussmanntransformation. MSC2000: 60G18,60H05,60H20. 1Introduction ThefractionalBrownianmotion(fBm)B={B t ,t≥0}ofHurstindexH∈(0,1)isacentered andcontinuousGaussianprocessverifyingB 0 =0a.s.and E[(B t −B s ) 2 ]=|t−s| 2H (1) foralls,t≥0.ObservethatB 1/2 isnothingbutstandardBrownianmotion.Equality(1)implies thatthetrajectoriesofBare(H−ε)-Ho¨ldercontinuous,foranyε>0smallenough.AsthefBm isselfsimilar(ofindexH)andhasstationaryincrements,itisusedasamodelinmanyfields (forexample,inhydrology,economics,financialmathematics,etc.).Inparticular,thestudyof stochasticdifferentialequations(SDEs)drivenbyafBmisimportantinviewoftheapplications. But,beforeraisingthequestionofexistenceand/oruniquenessforthistypeofSDEs,thefirst difficultyistogiveameaningtotheintegralwithrespecttoafBm.Itisindeedwell-knownthatB isnotasemimartingalewhenH6=1/2.Thus,theItoˆorStratonovichcalculusdoesnotapplyto thiscase.ThereareseveralwaysofbuildinganintegralwithrespecttothefBmandofobtaining achangeofvariablesformula.Letuspointoutsomeofthesecontributions: 1.Regularizationordiscretizationtechniques.Since1993,RussoandVallois[31]havedeveloped aregularizationprocedure,whosephilosophyissimilartothediscretization.Theyintroduce forward(generalizingItoˆ),backward,symmetric(generalizingStratonovich,seeDefinition3 below)stochasticintegralsandageneralizedquadraticvariation.Theregularization,ordis- cretizationtechnique,forfBmandrelatedprocesseshavebeenperformedby[12,17,32,36], inthecaseofzeroquadraticvariation(correspondingtoH>1/2).NotealsothatYounginte- grals[35],whichareoftenusedinthiscase,coincidewiththeforwardintegral(butalsowith thebackwardorsymmetricones,sincecovariationbetweenintegrandandintegratorisalways zero).Whentheintegratorhaspathswithfinitep-variationforp>2,forwardandbackward