It is well-known since the work of Pardoux and Peng [12] 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, see also
Pardoux and Rascanu [14]. This method was applied to the class of PDEs with a
nonlinear Neumann boundary condition first by Pardoux and Zhang [15]. 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 in [15] is a
Corollary of our main results
