## The Annals of Probability

### 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 . 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 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.

#### Article information

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

Dates
First available in Project Euclid: 5 September 2007

https://projecteuclid.org/euclid.aop/1189000935

Digital Object Identifier
doi:10.1214/009117906000001169

Mathematical Reviews number (MathSciNet)
MR2349582

Zentralblatt MATH identifier
1208.60033

#### Citation

Darses, Sébastien; Nourdin, Ivan. Stochastic derivatives for fractional diffusions. Ann. Probab. 35 (2007), no. 5, 1998--2020. doi:10.1214/009117906000001169. https://projecteuclid.org/euclid.aop/1189000935

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