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September 2007 Stochastic derivatives for fractional diffusions
Sébastien Darses, Ivan Nourdin
Ann. Probab. 35(5): 1998-2020 (September 2007). DOI: 10.1214/009117906000001169


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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Sébastien Darses. Ivan Nourdin. "Stochastic derivatives for fractional diffusions." Ann. Probab. 35 (5) 1998 - 2020, September 2007.


Published: September 2007
First available in Project Euclid: 5 September 2007

zbMATH: 1208.60033
MathSciNet: MR2349582
Digital Object Identifier: 10.1214/009117906000001169

Primary: 60G07 , 60G15
Secondary: 60G17 , 60H07

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

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 5 • September 2007
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