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

We will focus — in dimension one — on the SDEs of the type dX t = σ(X t )dB t + b(X t )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 synmetric integral.

[1]  Ivan Nourdin.  Calcul stochastique généralisé et applications au mouvement brownien fractionnaire : Estimation non paramétrique de la volatilité et test d'adéquation , 2004 .

[2]  A. Neuenkirch.  Optimal Approximation of Stochastic Differential Equations with Additive Fractional Noise , 2006 .

[3]  Antoine Lejay,et al.  An Introduction to Rough Paths , 2003 .

[4]  L. Young.  DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH SIGNALS ( I ) : AN EXTENSION OF AN INEQUALITY OF , 2004 .

[5]  David Nualart,et al.  Stochastic calculus with respect to the fractional Brownian motion and applications , 2003 .

[6]  M. Zähle.  Integration with respect to fractal functions and stochastic calculus. I , 1998 .

[7]  ORDINARY DIFFERENTIAL EQUATIONS WITH FRACTAL NOISE , 1999 .

[8]  Ivan Nourdin.  Schémas d'approximation associés à une équation différentielle dirigée par une fonction höldérienne; cas du mouvement brownien fractionnaire , 2005 .

[9]  P. Vallois,et al.  Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index H>=1/4 , 2003 .

[10]  D. Feyel,et al.  On Fractional Brownian Processes , 1999 .

[11]  Héctor J. Sussmann,et al.  An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point , 1977 .

[12]  Fabrice Baudoin,et al.  Étude en temps petit des solutions d'EDS conduites par des mouvements browniens fractionnaires , 2005 .

[13]  D. Feyel,et al.  CURVILINEAR INTEGRALS ALONG ENRICHED PATHS , 2006 .

[14]  P. Vallois,et al.  Stochastic calculus with respect to continuous finite quadratic variation processes , 2000 .

[15]  V. Bogachëv,et al.  Liens entre équations différentielles stochastiques et ordinaires en dimension infinie , 1995 .

[16]  I. Nourdin,et al.  On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion , 2006 .

[17]  D. Nualart,et al.  Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H∈(0,12) , 2005 .

[18]  O. Mazet,et al.  Stochastic calculus with respect to fractional Brownian motion , 2006 .

[19]  Intégrale stochastique pour le mouvement brownien fractionnaire , 2000 .

[20]  S. J. Lin,et al.  Stochastic analysis of fractional brownian motions , 1995 .

[21]  Ivan Nourdin,et al.  Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion , 2006 .

[22]  Giovanni Peccati,et al.  Central limit theorems for sequences of multiple stochastic integrals , 2005 .

[23]  I. Nourdin,et al.  Weighted power variations of fractional Brownian motion and application to approximating schemes , 2007 .

[24]  David Nualart,et al.  Stochastic Differential Equations with Additive Fractional Noise and Locally Unbounded Drift , 2003 .

[25]  A. Ruzmaikina.  Stochastic calculus with fractional Brownian motion , 1999 .

[26]  Francesco Russo,et al.  Forward, backward and symmetric stochastic integration , 1993 .

[27]  Denis Talay,et al.  Resolution trajectorielle et analyse numerique des equations differentielles stochastiques , 1983 .

[28]  D. Nualart,et al.  Differential equations driven by fractional Brownian motion , 2002 .

[29]  L. C. Young,et al.  An inequality of the Hölder type, connected with Stieltjes integration , 1936 .

[30]  Francesco Russo,et al.  m-order integrals and generalized Ito's formula; the case of a fractional Brownian motion with any Hurst index , 2005 .

[31]  Correcting Newton–Côtes integrals by Lévy areas , 2006, math/0601544.

[32]  Christian Bender.  An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter , 2003 .

[33]  Laure Coutin,et al.  Stochastic analysis, rough path analysis and fractional Brownian motions , 2002 .

[34]  A. S. ÜstünelÉcole.  Stochastic Analysis of the Fractional Brownian Motion , 1996 .

[35]  Francesco Russo,et al.  n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes , 2003 .