Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach

TL;DR

This paper derives the asymptotic expansion of the stationary distribution density for reflected Brownian motion in the quarter plane, extending analytic methods from discrete to continuous diffusion processes.

Contribution

It develops an analytic approach for diffusion processes with reflection, previously used mainly for discrete random walks, to analyze stationary distributions in the quarter plane.

Findings

Asymptotic expansion of stationary distribution density is obtained.

Main term of the expansion is identified based on parameters.

Method extends analytic techniques from discrete to continuous processes.

Abstract

Brownian motion in R 2 + with covariance matrix $\Sigma$ and drift $\mu$ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in R 2 + is found and its main term is identified depending on parameters ($\Sigma$, $\mu$, R). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in Z 2 + with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on R 2 + with reflections on the axes.