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Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

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  • Raluca M. Balan

    (University of Ottawa)

  • Ciprian A. Tudor

    (Université de Panthéon-Sorbonne Paris 1)

Abstract

We consider the stochastic heat equation with multiplicative noise $u_{t}=\frac{1}{2}\Delta u+u\dot{W}$ in ℝ+×ℝ d , whose solution is interpreted in the mild sense. The noise $\dot{W}$ is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α 1/2), respectively d

Suggested Citation

  • Raluca M. Balan & Ciprian A. Tudor, 2010. "Stochastic Heat Equation with Multiplicative Fractional-Colored Noise," Journal of Theoretical Probability, Springer, vol. 23(3), pages 834-870, September.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:3:d:10.1007_s10959-009-0237-3
    DOI: 10.1007/s10959-009-0237-3
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    References listed on IDEAS

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    1. Erhan Bayraktar & H. Vincent Poor & K. Ronnie Sircar, 2004. "Estimating The Fractal Dimension Of The S&P 500 Index Using Wavelet Analysis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(05), pages 615-643.
    2. Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    3. Coutin, Laure & Nualart, David & Tudor, Ciprian A., 2001. "Tanaka formula for the fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 94(2), pages 301-315, August.
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    Cited by:

    1. Zhang, Bin & Yao, Zhigang & Liu, Junfeng, 2023. "On a class of mixed stochastic heat equations driven by spatially homogeneous Gaussian noise," Statistics & Probability Letters, Elsevier, vol. 196(C).

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