On the stationary distribution of reflected Brownian motion in a non-convex wedge

TL;DR

This paper investigates the stationary distribution of reflected Brownian motion in non-convex wedges, providing a novel boundary value problem approach and identifying a family of stationary distributions applicable to both convex and non-convex cases.

Contribution

It introduces a new vector boundary value problem method for non-convex wedges and constructs a family of stationary distributions, extending the understanding beyond convex cases.

Findings

Stationary distribution characterized by a two-dimensional vector BVP.

Reduction to a single boundary condition in convex cases.

Existence of a one-parameter family of stationary distributions.

Abstract

We study the stationary reflected Brownian motion in a non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in the probabilistic literature. We prove that its stationary distribution can be found by solving a two dimensional vector boundary value problem (BVP) on a single curve for the associated Laplace transforms. The reduction to this kind of vector BVP seems to be new in the literature. As a matter of comparison, one single boundary condition is sufficient in the convex case. When the parameters of the model (drift, reflection angles and covariance matrix) are symmetric with respect to the bisector line of the cone, the model is reducible to a standard reflected Brownian motion in a convex cone. Finally, we construct a one-parameter family of distributions, which surprisingly provides, for any wedge (convex or not), one particular example of stationary distribution of a reflected Brownian motion.