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**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.

Reviewer: Rainer Buckdahn (Brest)

### 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.) |

### Keywords:

Backward stochastic differential equation; backward doubly stochastic differential equation; parabolic stochastic PDE; Neumann boundary condition
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\textit{B. Boufoussi} et al., Bernoulli 13, No. 2, 423--446 (2007; Zbl 1135.60038)

### References:

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[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 |

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