Stochastic derivatives for fractional diffusions

Abstract

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given $\sigma$-field $\mathcal{Q}$. In our framework, we recall well-known results about Markov--Wiener diffusions. We then focus mainly on the case where $X$ is a fractional diffusion and where $\mathcal{Q}$ is the past, the future or the present of $X$. We treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of $X$ when $X$ solves a stochastic differential equation driven by a fractional Brownian motion with Hurst index $H>1/2$. We give explicit formulas.Comment: Published in at http://dx.doi.org/10.1214/009117906000001169 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org