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Weak Solutions for SPDEs and Backward Doubly Stochastic Differential Equations

Author

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  • V. Bally

    (Université Paris VI)

  • A. Matoussi

    (Université du Maine)

Abstract

We give the probabilistic interpretation of the solutions in Sobolev spaces of parabolic semilinear stochastic PDEs in terms of Backward Doubly Stochastic Differential Equations. This is a generalization of the Feynman–Kac formula. We also discuss linear stochastic PDEs in which the terminal value and the coefficients are distributions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:1:d:10.1023_a:1007825232513
    DOI: 10.1023/A:1007825232513
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    References listed on IDEAS

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    1. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
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    Cited by:

    1. Qikang Ran & Tusheng Zhang, 2010. "Existence and Uniqueness of Bounded Weak Solutions of a Semilinear Parabolic PDE," Journal of Theoretical Probability, Springer, vol. 23(4), pages 951-971, December.
    2. 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).
    3. Qi Zhang & Huaizhong Zhao, 2012. "Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients," Journal of Theoretical Probability, Springer, vol. 25(2), pages 396-423, June.
    4. Tomasz Klimsiak, 2013. "On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators," Journal of Theoretical Probability, Springer, vol. 26(2), pages 437-473, June.
    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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