$On the absolute continuity of one dimensional SDE s driven by a fractional Brownian motion$

On the absolute continuity of one dimensional SDE's driven by a fractional Brownian motion

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On the absolute continuity of one-dimensional SDE's driven by a fractional Brownian motion Ivan Nourdin Universite Henri Poincare, Institut de Mathematiques Elie Cartan, B.P. 239 54506 Vandœuvre-les-Nancy Cedex, France Thomas Simon Universite d'Evry-Val d'Essonne, Equipe d'Analyse et Probabilites Boulevard Franc¸ois Mitterand, 91025 Evry Cedex, France Abstract The problem of absolute continuity for a class of SDE's driven by a real fractional Brownian motion of any Hurst index is adressed. First, we give an elementary proof of the fact that the solution to the SDE has a positive density for all t > 0 when the diffusion coefficient does not vanish, echoing in the fractional Brownian framework the main result we had previously obtained for Marcus equations driven by Levy processes [9]. Second, we extend in our setting the classical entrance-time criterion of Bouleau-Hirsch[2]. Keywords: Absolute continuity - Doss-Sussmann transformation - Fractional Brownian motion - Newton-Cotes SDE. MSC 2000: 60G18, 60H10. 1 Introduction In this note we study the absolute continuity of the solutions at any time t > 0 to SDE's of the type: Xt = x0 + ∫ t 0 b(Xs) ds + ∫ t 0 ?(Xs) dB H s , (1) where b, ? are real functions and BH is a linear fractional

doss

doss- sussmann transformation

brownian motion

transformation - fractional brownian

called newton-cotes

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On the absolute SDE’s driven by

continuity of one-dimensional a fractional Brownian motion

Ivan Nourdin ´ Universite´HenriPoincare´,InstitutdeMathe´matiquesElieCartan,B.P.239 54506Vandœuvre-l`es-NancyCe´dex,France Ivan.Nourdin@iecn.u-nancy.fr

Thomas Simon ´ ´ Universite´d’Evry-Vald’Essonne,Equiped’AnalyseetProbabilit´es ´ BoulevardFranc¸oisMitterand,91025EvryCe´dex,France Thomas.Simon@maths.univ-evry.fr

Abstract The problem of absolute continuity for a class of SDE’s driven by a real fractional Brownian motion of any Hurst index is adressed. First, we give an elementary proof of the fact that the solution to the SDE has a positive density for allt >0 when the diﬀusion coeﬃcient does not vanish, echoing in the fractional Brownian framework themainresultwehadpreviouslyobtainedforMarcusequationsdrivenbyLe´vy processes [9]. Second, we extend in our setting the classical entrance-time criterion of Bouleau-Hirsch[2].

Keywords:Absolute continuity - Doss-Sussmann transformation - Fractional Brownian motion-Newton-CˆotesSDE. MSC 2000:60G18, 60H10.

Introduction

In this note we study the absolute continuity of the solutions at any timet >0 to SDE’s of the type: Z Z t t H σ(Xs)dB ,(1) Xt=x0+b(Xs)ds+s 0 0 H whereb, σare real functions andBis a linear fractional Brownian motion (fBm) with Hurst indexH∈(0,(1),1). In means a particular type of linear non-semimartingale integrators,theso-calledNewton-Cˆotesintegrator,whichwasrecentlyintroducedbyone of uset al.Roughly speaking,[7] [8]. is an operator deﬁned through a limiting pro-cedureinvolvingtheusualNewton-Coˆteslinearapproximator(whoseorderdependson H the roughness of the pathBn`ioitossousaRalawkcab-dpmoceddr),andaforwarolsiV-la [12]. This gives a reasonable class of solutions to (1) as soon asσWeis regular enough. refer to [7] and [8] for more details on this topic. 2 The main interest ofstiasdltsrﬁitaheiytIrˆtroedroum’ofsfla:if:R→Ris + regular enough andY: Ω×R→Ris a bounded variation process, then for everyt≥0 Z Z t t H0H H0H f(B , Yt) =f(0, Y0) +f(B , Ys)dB+f(B , Ys)dYs,(2) t x s s y s 0 0

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