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Backward doubly SDEs and semilinear stochastic PDEs in a convex domain

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  • Matoussi, Anis
  • Sabbagh, Wissal
  • Zhang, Tusheng

Abstract

This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDSDEs) in a convex domain D without any regularity conditions on the boundary. Moreover, using a stochastic flow approach a probabilistic interpretation for a system of reflected SPDEs in a domain is given via such RBDSDEs. The solution is expressed as a pair (u,ν) where u is a predictable continuous process which takes values in a Sobolev space and ν is a random regular measure. The bounded variation process K, the component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D. This bounded variation process determines the measure ν from a particular relation by using the inverse of the flow associated to the diffusion operator.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:9:p:2781-2815
    DOI: 10.1016/j.spa.2016.12.010
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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 I," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 181-204, June.
    2. Xu, Tiange & Zhang, Tusheng, 2009. "White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3453-3470, October.
    3. Zhang, Tusheng, 2011. "Systems of stochastic partial differential equations with reflection: Existence and uniqueness," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1356-1372, June.
    4. Gyöngy, István & Rovira, Carles, 2000. "On Lp-solutions of semilinear stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 83-108, November.
    5. Hamadène, Said & Zhang, Jianfeng, 2010. "Switching problem and related system of reflected backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 403-426, April.
    6. 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. Yang, Xue & Zhang, Qi & Zhang, Tusheng, 2020. "Reflected backward stochastic partial differential equations in a convex domain," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6038-6063.
    2. Yang, Xue, 2019. "Reflected backward stochastic partial differential equations with jumps in a convex domain," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 126-136.
    3. Liu, Ruoyang & Tang, Shanjian, 2024. "The obstacle problem for stochastic porous media equations," Stochastic Processes and their Applications, Elsevier, vol. 167(C).

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