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

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