Penalization of Reflected SDEs and Neumann Problems of HJB Equations

TL;DR

This paper investigates penalization methods for reflected stochastic differential equations in complex domains and applies these results to establish existence, comparison, and maximum principles for Neumann boundary value problems of nonlinear PDEs with viscosity solutions.

Contribution

It introduces a penalization approximation approach for reflected SDEs in non-smooth, non-convex domains and applies this to prove key properties of associated PDEs with Neumann boundary conditions.

Findings

Convergence of penalized SDE solutions to reflected SDEs in uniform topology.

Existence and comparison principles for viscosity solutions of nonlinear PDEs with Neumann conditions.

Maximum principle established for linear PDEs with Neumann boundary conditions.

Abstract

In this paper we first study the penalization approximation of stochastic differential equations reflected in a domain which satisfies conditions (A) and (B) and prove that the sequence of solutions of the penalizing equations converges in the uniform topology to the solution of the corresponding reflected stochastic differential equation. Then by using this convergence result, we consider partial differential equations with Neumann boundary conditions in domains neither smooth nor convex and prove the existence and comparison principle of viscosity solutions of such nonlinear PDEs. Also, by applying the support of reflected diffusions established in \cite{ren-wuAP}, we establish the maximum principle for the viscosity solutions of linear PDEs with Neumann boundary conditions.