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−Ltu(t,x)+∂φ(u(t,x))∋f(t,x,u(t,x),(∇uσ)(t,x)),t>0,x∈D,∂u(t,x)∂n+∂ψ(u(t,x))∋g(t,x,u(t,x)),t>0,x∈Bd(D),u(0,x)=h(x),x∈D¯, where ∂φ∂φ and ∂ψ∂ψ are subdifferential operators and LtLt is a second-differential operator given by Ltv(x)=12∑i,j=1d(σσ∗)ij(t,x)∂2v(x)∂xi∂xj+∑i=1dbi(t,x)∂v(x)∂xi. The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality: {dYt+F(t,Yt,Zt)dt+G(t,Yt)dAt∈∂φ(Yt)dt+∂ψ(Yt)dAt+ZtdWt,0≤t≤T,YT=ξ, where (At)t≥0(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.