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BSDEs generated by fractional space-time noise and related SPDEs

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  • Hu, Yaozhong
  • Li, Juan
  • Mi, Chao

Abstract

This paper is concerned with the backward stochastic differential equations whose generator is a weighted fractional Brownian field: Yt=ξ+∫tTYsW(ds,Bs)−∫tTZsdBs, 0≤t≤T, where W is a (d+1)-parameter weighted fractional Brownian field of Hurst parameter H=(H0,H1,⋯,Hd), which provide probabilistic interpretations (Feynman-Kac formulas) for certain linear stochastic partial differential equations with colored space-time noise. Conditions on the Hurst parameter H and on the decay rate of the weight are given to ensure the existence and uniqueness of the solution pair. Moreover, the explicit expression for both components Y and Z of the solution pair is given.

Suggested Citation

  • Hu, Yaozhong & Li, Juan & Mi, Chao, 2023. "BSDEs generated by fractional space-time noise and related SPDEs," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    • Handle: RePEc:eee:apmaco:v:450:y:2023:i:c:s0096300323001480
    DOI: 10.1016/j.amc.2023.127979
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    References listed on IDEAS

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    1. Buckdahn, Rainer & Ma, Jin, 2001. "Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part II," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 205-228, June.
    2. Buckdahn, Rainer & Ma, Jin, 2001. "Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 181-204, June.
    3. Hu, Yaozhong & Nualart, David & Song, Jian, 2013. "A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1083-1103.
    4. V. Bally & A. Matoussi, 2001. "Weak Solutions for SPDEs and Backward Doubly Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 14(1), pages 125-164, January.
    5. Anis Matoussi & Michael Scheutzow, 2002. "Stochastic PDEs Driven by Nonlinear Noise and Backward Doubly SDEs," Journal of Theoretical Probability, Springer, vol. 15(1), pages 1-39, January.
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