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

TL;DR

This paper proves the continuity of the extended Feynman-Kac formula for a broad class of parabolic equations with nonlinear boundary conditions involving convex functions, addressing gaps in previous proofs.

Contribution

It establishes the continuity of the Feynman-Kac formula for generalized parabolic equations with nonlinear boundary conditions involving multivalued gradients, extending prior results.

Findings

Proved continuity of the Feynman-Kac formula in a generalized setting.

Extended the class of PDEs where probabilistic representations are valid.

Addressed and corrected gaps in earlier proofs of continuity.

Abstract

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.