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

Abstract

In this article we prove the continuity of the deterministic function $u:[0,T]\times \mathcal{\bar{D}}\rightarrow \mathbb{R}$, 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 $(X_{s}^{t,x},A_{s}^{t,x})_{s\geq t}$ is the solution of a stochastic differential equation with reflecting boundary conditions.