Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

Backward stochastic differential equations (BSDEs) were introduced by Pardoux and Peng [6], and it was shown in various papers that stochastic differential equations (SDEs) of this type give a probabilistic representation for the solution (at least in the viscosity sense) of a large class of system of semi-linear parabolic partial differential equations (PDEs). A new class of BSDEs, called backward doubly stochastic (BDSDEs), was considered by Pardoux and Peng [7]. This new kind of BSDEs seems to be suitable giving a probabilistic representation for a system of parabolic stochastic partial differential equations (SPDEs). We refer to Pardoux and Peng [7] for the link between SPDEs and BDSDEs in the particular case where solutions of SPDEs are regular. The more general situation is much more delicate to treat because of difficulties of extending the notion of stochastic viscosity solutions to SPDEs. The notion of viscosity solution for PDEs was introduced by Crandall and Lions [3] for certain first-order Hamilton–Jacobi equations. Today the theory has become an important tool in many applied fields, especially in optimal control theory and numerous subjects related to it.