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Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise

Author

Listed:
  • Duncan, T.E.
  • Maslowski, B.
  • Pasik-Duncan, B.

Abstract

In this paper, some explicit solutions are given for stochastic differential equations in a Hilbert space with a multiplicative fractional Gaussian noise. This noise is the formal derivative of a fractional Brownian motion with the Hurst parameter in the interval (1/2,1). These solutions can be weak, strong or mild depending on the specific assumptions. The problem of stochastic stability of these equations is considered and for various notions of stability, sufficient conditions are given for stability. The noise may stabilize or destabilize the corresponding deterministic solutions. Various examples of stochastic partial differential equations are given that satisfy the assumptions for explicit solutions or stability.

Suggested Citation

  • Duncan, T.E. & Maslowski, B. & Pasik-Duncan, B., 2005. "Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 115(8), pages 1357-1383, August.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:8:p:1357-1383
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    References listed on IDEAS

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    1. Peszat, Szymon & Zabczyk, Jerzy, 1997. "Stochastic evolution equations with a spatially homogeneous Wiener process," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 187-204, December.
    2. Grecksch, W. & Anh, V. V., 1999. "A parabolic stochastic differential equation with fractional Brownian motion input," Statistics & Probability Letters, Elsevier, vol. 41(4), pages 337-346, February.
    3. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Lihong Guo, 2024. "Renormalization Group Method for a Stochastic Differential Equation with Multiplicative Fractional White Noise," Mathematics, MDPI, vol. 12(3), pages 1-20, January.
    2. Zhang, Yinghan & Yang, Xiaoyuan, 2015. "Fractional stochastic Volterra equation perturbed by fractional Brownian motion," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 20-36.
    3. Barbu, Viorel & Brzeźniak, Zdzisław & Hausenblas, Erika & Tubaro, Luciano, 2013. "Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 934-951.
    4. Boufoussi, Brahim & Hajji, Salah, 2017. "Stochastic delay differential equations in a Hilbert space driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 222-229.
    5. Issoglio, E. & Riedle, M., 2014. "Cylindrical fractional Brownian motion in Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3507-3534.
    6. Fan, Xiliang & Yuan, Chenggui, 2016. "Lyapunov exponents of PDEs driven by fractional noise with Markovian switching," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 39-50.

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