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

In this article we prove the continuity of the deterministic function $u:[0,T]\times \bar{\mathcal{D}}\to \mathbb{R}$, defined by $u(t,x):=Y_t^{t,x}$, where the process $(Y_s^{t,x}{)}_{s\in [t,T]}$ is given by the generalized multivalued backward stochastic differential equation: \begin{equation*} \left{ \begin{array}{l} -dY_{s}^{t,x}+\partial \varphi(Y_{s}^{t,x})ds+\partial\psi(Y_{s}^{t,x})dA_{s}^{t,x}\ni 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\leq s < T, \ {Y_{T}=h(X_{T}^{t,x}).} \end{array} \right. \end{equation*} The process $(X_s^{t,x},A_s^{t,x}{)}_{s\ge t}$ is the solution of a stochastic differential equation with reflecting boundary conditions.