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

Ivan Nourdin (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this paper, we will focus - in dimension one - on the SDEs of the type dX_t=s(X_t)dB_t+b(X_t)dt where B is a fractional Brownian motion. Our principal motivation is to describe one of the simplest theory - from our point of view - allowing to study this SDE, and this for any Hurst index H between 0 and 1. We will consider several definitions of solution and we will study, for each one of them, in which condition one has existence and uniqueness. Finally, we will examine the convergence or not of the canonical scheme associated to our SDE, when the integral with respect to fBm is defined using the Russo-Vallois symmetric integral.

Mots clés: Stochastic differential equation; fractional Brownian motion; Russo-Vallois integrals; Newton-Côtes integral; approximation schemes; Doss-Sussmann transformation

Domaines: Probabilités [math.PR]

Ivan Nourdin. A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. 2007. ⟨hal-00012961v5⟩