Stochastic derivatives for fractional diffusions

Darses, Sébastien ; Nourdin, Ivan

Ann. Probab., Tome 35 (2007) no. 1, p. 1998-2020 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2007-09-14
Classification:  Stochastic derivatives,  Nelson’s derivative,  fractional Brownian motion,  fractional differential equation,  Malliavin calculus,  60G07,  60G15,  60G17,  60H07