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: { ∂ u ( t , x ) ∂ t − L t u ( t , x ) + ∂ φ ( u ( t , x ) ) ∋ f ( t , x , u ( t , x ) , ( ∇ u σ ) ( t , x ) ) , t > 0 , x ∈ D , ∂ u ( t , x ) ∂ n + ∂ ψ ( u ( t , x ) ) ∋ g ( t , x , u ( t , x ) ) , t > 0 , x ∈ B d ( D ) , u ( 0 , x ) = h ( x ) , x ∈ D ¯ , where ∂ φ and ∂ ψ are subdifferential operators and L t is a second-differential operator given by L t v ( x ) = 1 2 ∑ i , j = 1 d ( σ σ ∗ ) i j ( t , x ) ∂ 2 v ( x ) ∂ x i ∂ x j + ∑ i = 1 d b i ( t , x ) ∂ v ( x ) ∂ x i . The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality: { d Y t + F ( t , Y t , Z t ) d t + G ( t , Y t ) d A t ∈ ∂ φ ( Y t ) d t + ∂ ψ ( Y t ) d A t + Z t d W t , 0 ≤ t ≤ T , Y T = ξ , where ( A t ) 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.