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Pathwise definition of second-order SDEs

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  • Quer-Sardanyons, Lluís
  • Tindel, Samy

Abstract

In this article, a class of second-order differential equations on [0,1], driven by a γ-Hölder continuous function for any value of γ∈(0,1) and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks to Young integration techniques. We then study the differentiability of the solution with respect to the driving process and consider the case where the equation is driven by a fractional Brownian motion, with two aims in mind: show that the solution that we have produced coincides with the one which would be obtained with Malliavin calculus tools, and prove that the law of the solution is absolutely continuous with respect to the Lebesgue measure.

Suggested Citation

  • Quer-Sardanyons, Lluís & Tindel, Samy, 2012. "Pathwise definition of second-order SDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 466-497.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:2:p:466-497
    DOI: 10.1016/j.spa.2011.08.014
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    References listed on IDEAS

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    1. Nualart, David & Pardoux, Etienne, 1991. "Second order stochastic differential equations with Dirichlet boundary conditions," Stochastic Processes and their Applications, Elsevier, vol. 39(1), pages 1-24, October.
    2. 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.
    3. 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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