A stochastic approach to a multivalued Dirichlet-Neumann problem

We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality (PVI) with a mixed nonlinear multivalued Neumann-Dirichlet boundary condition: where [not partial differential][phi] and [not partial differential][psi] are subdifferential operators and is a second-differential operator given by The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality: where (At)t>=0 is a continuous one-dimensional increasing measurable process, and then we obtain a Feynman-Kaç representation formula for the viscosity solution of the PVI problem.