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 σ-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.