Penalty Method for Obliquely Reflected Diffusions

TL;DR

This paper develops a general penalty method to approximate multidimensional obliquely reflected diffusions by stochastic differential equations without reflection, providing broad conditions for the approximation's validity.

Contribution

It introduces general sufficient conditions for the penalty approximation of obliquely reflected diffusions, extending beyond specific sequences used previously.

Findings

Established broad conditions for penalty approximation validity

Unified approach applicable to various reflection directions

Enhanced theoretical understanding of reflected diffusion approximations

Abstract

Take a multidimensional normally or obliquely reflected diffusion in a smooth domain. Approximate it by solutions of stochastic differential equations without reflection using the penalty method. That is, we approximate the reflection term with an additional drift term. In the existing literature, usually a specific approximating sequence is provided in order to prove existence of a reflected diffusion. In this article, we provide general sufficient conditions on the approximating coefficients.