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

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  • Buckdahn, Rainer
  • Ma, Jin

Abstract

This paper is a continuation of our previous work (Part I, Stochastic Process. Appl. 93 (2001) 181-204), with the main purpose of establishing the uniqueness of the stochastic viscosity solution introduced there. We shall prove a comparison theorem between a stochastic viscosity solution and an [omega]-wise stochastic viscosity solution, which will lead to the uniqueness results. As the byproducts we extend the measurable section theorem of Dellacherie and Meyer (1978), and a fundamental lemma of Crandall et al. (1992)

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  • 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.
    • Handle: RePEc:eee:spapps:v:93:y:2001:i:2:p:205-228
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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. 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.
    8. 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.
    9. 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.
    10. 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).
    11. 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.
    12. 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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