Penalization method for a nonlinear Neumann PDE via weak solutions of reflected SDEs

TL;DR

This paper introduces a probabilistic penalization approach to approximate viscosity solutions of nonlinear PDEs with Neumann boundary conditions, using reflected SDEs and backward stochastic equations.

Contribution

It provides a novel probabilistic framework for approximating solutions of nonlinear PDEs with Neumann boundary conditions via penalized reflected SDEs.

Findings

Proved weak uniqueness of reflected SDE solutions.

Established convergence of penalized SDE solutions to the PDE solution.

Demonstrated the probabilistic representation of the viscosity solution.

Abstract

In this paper we prove an approximation result for the viscosity solution of a system of semi-linear partial differential equations with continuous coefficients and nonlinear Neumann boundary condition. The approximation we use is based on a penalization method and our approach is probabilistic. We prove the weak uniqueness of the solution for the reflected stochastic differential equation and we approximate it (in law) by a sequence of solutions of stochastic differential equations with penalized terms. Using then a suitable generalized backward stochastic differential equation and the uniqueness of the reflected stochastic differential equation, we prove the existence of a continuous function, given by a probabilistic representation, which is a viscosity solution of the considered partial differential equation. In addition, this solution is approximated by solutions of penalized partial differential equations.