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Malliavin Calculus for Fractional Delay Equations

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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.

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Authors and Affiliations

  1. Depto. de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 07000, Mexico, DF, Mexico

    Jorge A. León

  2. Institut Élie Cartan Nancy, B.P. 239, 54506, Vandoeuvre-lès-Nancy Cedex, France

    Samy Tindel

Authors
  1. Jorge A. León
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  2. Samy Tindel
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Corresponding author

Correspondence to Jorge A. León.

Additional information

J.A. León is partially supported by the CONACyT grant 98998. S. Tindel is partially supported by the ANR grant ECRU.

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León, J.A., Tindel, S. Malliavin Calculus for Fractional Delay Equations. J Theor Probab 25, 854–889 (2012). https://doi.org/10.1007/s10959-011-0349-4

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  • DOI: https://doi.org/10.1007/s10959-011-0349-4

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