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

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.

21 pages

References in corpus (1)

• Penalization methods for the Skorokhod problem and reflecting SDEs with jumps

Cited by in corpus (4)

• On the continuity of the probabilistic representation of a semilinear Neumann-Dirichlet problem
• BSDEs driven by $|z{|}^2/y$ and applications to PDEs and decision theory
• Probabilistic representation of parabolic stochastic variational inequality with Dirichlet-Neumann boundary and variational generalized backward doubly stochastic differential equations
• Penalization of Reflected SDEs and Neumann Problems of HJB Equations