Stochastic derivatives for fractional diffusions

Sébastien Darses (1) , Ivan Nourdin (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given sigma-field Q. In our framework, we recall well known results about Markov Wiener diffusions. We afterwards mainly focus on the case where X is a fractional diffusion and where 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.

Mots clés: stochastic derivatives; Nelson's derivative; fractional Brownian motion; fractional differential equation; Malliavin calculus

Domaines: Probabilités [math.PR]

Sébastien Darses, Ivan Nourdin. Stochastic derivatives for fractional diffusions. 2006. ⟨hal-00022829v3⟩