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Malliavin Calculus for Fractional Delay Equations

Author

Listed:
  • Jorge A. León

    (CINVESTAV-IPN)

  • Samy Tindel

    (Institut Élie Cartan Nancy)

Abstract

In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a C ∞-density. To this purpose, we use Malliavin calculus based on the Fréchet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.

Suggested Citation

  • Jorge A. León & Samy Tindel, 2012. "Malliavin Calculus for Fractional Delay Equations," Journal of Theoretical Probability, Springer, vol. 25(3), pages 854-889, September.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:3:d:10.1007_s10959-011-0349-4
    DOI: 10.1007/s10959-011-0349-4
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    References listed on IDEAS

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    1. Nualart, David & Saussereau, Bruno, 2009. "Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 391-409, February.
    2. Quer-Sardanyons, Lluís & Tindel, Samy, 2007. "The 1-d stochastic wave equation driven by a fractional Brownian sheet," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1448-1472, October.
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