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

Abstract

Brownian motion in $\mathbf{R}_{+}^{2}$ 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 $\mathbf{R}_{+}^{2}$ 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 $\mathbf{Z}_{+}^{2}$ with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on $\mathbf{R}_{+}^{2}$ with reflections on the axes.