Continuity of the Feynman-Kac formula for a generalized parabolic equation

It is well-known that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman-Kac formula to semilinear PDEs. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang in 1998. However, the proof of continuity of the extended Feynman-Kac formula with respect to x (resp. to (t,x)) is not correct in that paper. Here we consider a more general situation, where both the equation and the boundary condition involve the (possibly multivalued) gradient of a convex function. We prove the required continuity. The result for the class of equations studied Pardoux and Zhang in their paper from 1998, as well as those considered by Maticiuc and Rcanu in their paper from 2010, are Corollaries of our main results.