Stochastic derivatives for fractional diffusions

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given a-field 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 62 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.