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A quasilinear stochastic partial differential equation driven by fractional white noise

Author

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  • Grecksch Wilfried

    (Martin-Luther-University of Halle-Wittenberg, Department for Mathematics and Computer Sciences, Institute for Optimization and Stochastics, 06099 Halle (Saale), Germany.)

  • Roth Christian

    (Martin-Luther-University of Halle-Wittenberg, Department for Mathematics and Computer Sciences, Institute for Optimization and Stochastics, 06099 Halle (Saale), Germany. Email: christian.roth@mathematik.uni-halle.de)

Abstract

The objective of the paper is to give the representation of a solution of a quasilinear stochastic partial differential equation driven by scalar fractional Brownian motion B H(t), H ∈ (½, 1), in the white noise framework for fractional Brownian motion. The solution is represented as a Wick product between a fractional Wick exponential and the solution of a path wise deterministic parabolic partial differential equation. Thereby a fractional theory of fractional translation operators is developed and used in the spirit of Benth and Gjessing [F. E. Benth and H. Gjessing. A nonlinear parabolic equation with noise. Potential Analysis 12 (2000), 385–401] who used it in the pure Brownian motion case.

Suggested Citation

  • Grecksch Wilfried & Roth Christian, 2008. "A quasilinear stochastic partial differential equation driven by fractional white noise," Monte Carlo Methods and Applications, De Gruyter, vol. 13(5-6), pages 353-367, January.
    • Handle: RePEc:bpj:mcmeap:v:13:y:2008:i:5-6:p:353-367:n:2
    DOI: 10.1515/mcma.2007.019
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    References listed on IDEAS

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    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
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