On the continuity of the probabilistic representation of a semilinear Neumann-Dirichlet problem

In this article we prove the continuity of the deterministic function u:[0,T]×D̄→R, defined by u(t,x):=Ytt,x, where the process (Yst,x)s∈[t,T] is given by the generalized multivalued backward stochastic differential equation:{−dYst,x+∂φ(Yst,x)ds+∂ψ(Yst,x)dAst,x∋f(s,Xst,x,Yst,x)ds+g(s,Xst,x,Yst,x)dAst,x−Zst,xdWs,t≤s