Stochastic Systems

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

Sandro Franceschi and Irina Kourkova

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ssy/1495785617

Digital Object Identifier
doi:10.1214/16-SSY218

Mathematical Reviews number (MathSciNet)
MR3663338

Zentralblatt MATH identifier
1365.60072

Subjects
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]

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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. https://projecteuclid.org/euclid.ssy/1495785617


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