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] x (D) over bar -> R, defined by u (t, x) := Y-t(t,x), where the process (Y-s(t,x))(s is an element of[t,T]) is given by the generalized multivalued backward stochastic differential equation: {-dY(s)(t,x) + partial derivative phi(Y-s(t,x))ds + partial derivative psi(Y-s(t,x))dA(s)(t,x) (sic) f(s, X-s(t,x), Y-s(t,x))ds +g(s, X-s(t,x), Y-s(t,x))dA(s)(t,x) - Z(s)(t,x) dW(s), t <= s < T Y-T = h(X-T(t,x)). The process (X-s(t,x), A(s)(t,x))(s >= t) is the solution of a stochastic differential equation with reflecting boundary conditions. (C) 2015 Elsevier B.V. All rights reserved.