Stochastic Systems

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

Sandro Franceschi and Irina Kourkova

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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.

Article information

Stoch. Syst., Volume 7, Number 1 (2017), 32-94.

Received: April 2016
First available in Project Euclid: 26 May 2017

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 37L40: Invariant measures
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

Reflected Brownian motion in the quarter plane stationary distribution Laplace transform asymptotic analysis saddle-point method Riemann surface

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Franceschi, Sandro; Kourkova, Irina. Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach. Stoch. Syst. 7 (2017), no. 1, 32--94. doi:10.1214/16-SSY218.

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