Abstract
We study the following quasilinear partial differential equation with two subdifferential operators:where for and The operator (resp. ) is the subdifferential of the convex lower semicontinuous function (resp. ). We define the viscosity solution for such kind of partial differential equation and prove the uniqueness of the viscosity solution when does not depend on . To prove the existence of a viscosity solution, a stochastic representation formula of Feymann–Kac type will be developed. For this end, we investigate a fully coupled forward–backward stochastic variational inequality.
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