×

Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions. (English) Zbl 1135.60038

The authors’ objective in the present paper is to give a stochastic interpretation to a semilinear parabolic stochastic partial differential equation (SPDE) over a bounded domain, endowed with a semilinear Neumann boundary condition. In their pioneering paper [Probab. Theory Relat. Fields 98, No. 2, 209–227 (1994; Zbl 0792.60050)], E. Pardoux and S. Peng introduced backward doubly stochastic differential equations (BDSDEs) and proved, in the case of smooth coefficients, that their solution describes the classical solution of associated parabolic SPDEs. However, in their works on backward stochastic differential equations (BSDEs)E. Pardoux and S. G. Peng [Syst. Control Lett. 14 , No. 1, 55–61 (1990; Zbl 0692.93064); Lect. Notes Control Inf. Sci. 176, 200–217 (1992; Zbl 0766.60079)] showed that the solution of a BSDE with only Lipschitz assumptions on the coefficients provides the unique viscosity solution of the associated (deterministic) PDE. This has been the starting point for several recent works which have the objective to get the same generality of results concerning the link between SPDEs and BDSDEs. Inspired by the ideas of P. L. Lions and P. E. Souganidis, R. Buckdahn and J. Ma introduced for semilinear parabolic SPDEs whose diffusion coefficient depends only on the solution but not on its gradient a notion of viscosity solution, and they proved its existence and uniqueness with the help of an appropriate Doss-Sussman transformation. Referring to this approach, the authors of the present paper describe the stochastic viscosity solution of a semilinear parabolic SPDE with a semilinear Neumann boundary condition with the help of an associated BDSDE. The authors’ work can be considered as generalization of an earlier paper by E. Pardoux and S. Zhang [Probab. Theory Relat. Fields 110, No. 4, 535–558 (1998, Zbl 0909.60046)] in which the link between PDEs with nonlinear Neumann boundary condition and BSDEs was studied.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Boufoussi, B. and Van Casteren, J. (2004). An approximation result for a nonlinear Neumann boundary value problem via BSDEs., Stochastic Process. Appl. 114 331–350. · Zbl 1073.60062 · doi:10.1016/j.spa.2004.06.003
[2] Buckdahn, R. and Ma, J. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations I., Stochastic Process. Appl. 93 181–204. · Zbl 1053.60065 · doi:10.1016/S0304-4149(00)00093-4
[3] Crandall, M.G. and Lions, P.L. (1983). Viscosity solutions of Hamilton–Jacobi equations., Trans. Amer. Math. Soc. 277 1–42. · Zbl 0599.35024 · doi:10.2307/1999343
[4] Lions, P.L. and Souganidis, P.E. (1998). Fully nonlinear stochastic partial differential eqution., C. R. Acad. Sci. Paris, Sér. I Math. 326 1085–1092. · Zbl 1002.60552 · doi:10.1016/S0764-4442(98)80067-0
[5] Lions, P.L. and Sznitman, A.S. (1984). Stochastic differential equation with reflecting boundary conditions, Comm. Pure Appl. Math. 37 511–537. · Zbl 0598.60060 · doi:10.1002/cpa.3160370408
[6] Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation., Syst. Control Lett. 14 55–61. · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[7] Pardoux, E. and Peng, S. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs., Probab. Theory Related Fields 98 209–227. · Zbl 0792.60050 · doi:10.1007/BF01192514
[8] Pardoux, E. and Zhang, S. (1998). Generalized BSDEs and nonlinear boundary value problems., Probab. Theory Related Fields 110 535–558. · Zbl 0909.60046 · doi:10.1007/s004400050158
[9] Protter, P. (1990)., Stochastic Integration and Differential Equations: A New Approach . Berlin: Springer-Verlag. · Zbl 0694.60047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.