Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations

Abstract

We derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with sub-differential of a lower semi-continuous convex function under a non Lipschitz condition. This study allows us give a probabilistic representation (in stochastic viscosity sense) to the parabolic variational stochastic partial differential equations with Dirichlet-Neumann conditions.

This version is 39 pages long and has been accepted to Stochastics who will publish the final version

References in corpus (4)

• Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions
• Stochastic approach for a multivalued Dirichlet-Neumann problem
• On the continuity of the probabilistic representation of a semilinear Neumann-Dirichlet problem
• Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs