The Annals of Probability

Stochastic derivatives for fractional diffusions

Sébastien Darses and Ivan Nourdin

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 5 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1189000935

Digital Object Identifier
doi:10.1214/009117906000001169

Mathematical Reviews number (MathSciNet)
MR2349582

Zentralblatt MATH identifier
1208.60033

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

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

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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References

  • Baudoin, F. and Nualart, D. (2003). Equivalence of Volterra processes. Stochastic Process. Appl. 107 327--350.
  • Cresson, J. and Darses, S. (2006). Plongement stochastique des systèmes lagrangiens. C. R. Acad. Sci. Paris Ser. I 342 333--336.
  • Cresson, J. and Darses, S. (2006). Stochastic embedding of dynamical systems. Preprint I.H.É.S. 06/27.
  • Darses, S. and Saussereau, B. (2006). Time reversal for drifted fractional Brownian motion with Hurst index $H>1/2$. Preprint Besancon.
  • Doss, H. (1977). Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Probab. Statist. 13 99--125.
  • Föllmer, H. (1986). Time reversal on Wiener space. Stochastic Processes---Mathematics and Physics (Bielefeld, 1984) 119--129. Lecture Notes in Math. 1158. Springer, Berlin.
  • Guerra, J. and Nualart, D. (2005). The $1/H$-variation of the divergence integral with respect to the fractional Brownian motion for $H>1/2$ and fractional Bessel processes. Stochastic Process. Appl. 115 91--115.
  • Millet, A., Nualart, D. and Sanz, M. (1989). Integration by parts and time reversal for diffusion processes. Ann. Probab. 17 208--238.
  • Nelson, E. (1967). Dynamical Theories of Brownian Motion. Princeton Univ. Press, Princeton, NJ. (2nd ed. available at http://www.math.princeton.edu/~nelson/books/bmotion.pdf.)
  • Nourdin, I. and Simon, T. (2006). On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. Statist. Probab. Lett. 76 907--912.
  • Nualart, D. (1996). The Malliavin Calculus and Related Topics. Springer, New York.
  • Nualart, D. (2003). Stochastic calculus with respect to the fractional Brownian motion and applications. Contemp. Math. 336 3--39.
  • Nualart, D. and Ouknine, Y. (2002). Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102 103--116.
  • Pipiras, V. and Taqqu, M. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7 873--897.
  • Russo, F. and Vallois, P. (2007). Elements of stochastic calculus via regularisation. Séminaire de Probabilités. To appear.
  • Sussmann, H. J. (1977). An interpretation of stochastic differential equations as ordinary differential equations which depend on a sample point. Bull. Amer. Math. Soc. 83 296--298.
  • Thieullen, M. (1993). Second order stochastic differential equations and non-Gaussian reciprocal diffusions. Probab. Theory Related Fields 97 231--257.
  • Young, L. C. (1936). An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67 251--282.
  • Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 333--374.