Localization for constrained martingale problems and optimal conditions for uniqueness of reflecting diffusions in 2-dimensional domains

TL;DR

This paper establishes optimal conditions for the existence and uniqueness of reflecting diffusions in 2D domains with oblique reflection, extending previous results to include cusps and more general settings.

Contribution

It introduces a new localization technique for constrained martingale problems, broadening the scope of reflecting diffusion analysis in complex domains.

Findings

Proved existence and uniqueness under geometric conditions

Extended conditions to include cusps in domains

Established a new localization result for constrained martingale problems

Abstract

We prove existence and uniqueness for semimartingale reflecting diffusions in 2-dimensional piecewise smooth domains with varying, oblique directions of reflection on each "side", under geometric, easily verifiable conditions. Our conditions are optimal in the sense that, in the case of a convex polygon with constant direction of reflection on each side, they reduce to the conditions of Dai and Williams (1996), which are necessary for existence of Reflecting Brownian Motion. Moreover our conditions allow for cusps. Our argument is based on a new localization result for constrained martingale problems which holds quite generally: as an additional example, we show that it holds for diffusions with jump boundary conditions.