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Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

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  • Buckdahn, Rainer
  • Ma, Jin

Abstract

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205-228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss-Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a "point-wise" manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the [omega]-wise, "deterministic" viscosity solution to the PDE with random coefficients will be established.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:93:y:2001:i:2:p:181-204
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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.
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    Cited by:

    1. 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).
    2. Neeraj Bhauryal & Ana Bela Cruzeiro & Carlos Oliveira, 2024. "Pathwise Stochastic Control and a Class of Stochastic Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1967-1990, November.
    3. Marcel Nutz, 2011. "A Quasi-Sure Approach to the Control of Non-Markovian Stochastic Differential Equations," Papers 1106.3273, arXiv.org, revised May 2012.
    4. Francesco, MENONCIN, 2002. "Investment Strategies in Incomplete Markets : Sufficient Conditions for a Closed Form Solution," LIDAM Discussion Papers IRES 2002033, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    5. Matoussi, Anis & Sabbagh, Wissal & Zhang, Tusheng, 2017. "Backward doubly SDEs and semilinear stochastic PDEs in a convex domain," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2781-2815.
    6. Matoussi, A. & Piozin, L. & Popier, A., 2017. "Stochastic partial differential equations with singular terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 831-876.
    7. Aman, Auguste & Mrhardy, Naoul, 2013. "Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 863-874.
    8. Keller, Christian & Zhang, Jianfeng, 2016. "Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 735-766.
    9. Francesco, MENONCIN, 2002. "Investment Strategies for HARA Utility Function : A General Algebraic Approximated Solution," LIDAM Discussion Papers IRES 2002034, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    10. Matoussi Anis & Sabbagh Wissal, 2016. "Numerical computation for backward doubly SDEs with random terminal time," Monte Carlo Methods and Applications, De Gruyter, vol. 22(3), pages 229-258, September.
    11. Xanthi-Isidora Kartala & Nikolaos Englezos & Athanasios N. Yannacopoulos, 2020. "Future Expectations Modeling, Random Coefficient Forward–Backward Stochastic Differential Equations, and Stochastic Viscosity Solutions," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 403-433, May.

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    2. Neeraj Bhauryal & Ana Bela Cruzeiro & Carlos Oliveira, 2024. "Pathwise Stochastic Control and a Class of Stochastic Partial Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1967-1990, November.
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    8. Keller, Christian & Zhang, Jianfeng, 2016. "Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 735-766.
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