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.
Categories: math.PR