The Annals of Probability

Stochastic derivatives for fractional diffusions

Sébastien Darses and Ivan Nourdin

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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.

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Ann. Probab., Volume 35, Number 5 (2007), 1998-2020.

First available in Project Euclid: 5 September 2007

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Zentralblatt MATH identifier

Primary: 60G07: General theory of processes 60G15: Gaussian processes
Secondary: 60G17: Sample path properties 60H07: Stochastic calculus of variations and the Malliavin calculus

Stochastic derivatives Nelson’s derivative fractional Brownian motion fractional differential equation Malliavin calculus


Darses, Sébastien; Nourdin, Ivan. Stochastic derivatives for fractional diffusions. Ann. Probab. 35 (2007), no. 5, 1998--2020. doi:10.1214/009117906000001169.

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