On approximate continuity and the support of reflected stochastic differential equations

TL;DR

This paper establishes an approximate continuity result and characterizes the support of reflected stochastic differential equations in certain domains, advancing the understanding of their behavior and applications.

Contribution

It proves an approximate continuity theorem and characterizes the support of reflected diffusions in specific domains, completing the support theorem in the uniform topology.

Findings

Support theorem for reflected diffusions in uniform topology

Characterization of support in Hölder norm

Boundary-interior maximum principle for subharmonic functions

Abstract

In this paper we prove an approximate continuity result for stochastic differential equations with normal reflections in domains satisfying Saisho's conditions, which together with the Wong-Zakai approximation result completes the support theorem for such diffusions in the uniform convergence topology. Also by adapting Millet and Sanz-Sol\'{e}'s idea, we characterize in H\"{o}lder norm the support of diffusions reflected in domains satisfying the Lions-Sznitman conditions by proving limit theorems of adapted interpolations. Finally we apply the support theorem to establish a boundary-interior maximum principle for subharmonic functions.