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 Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach
 Sandro Franceschi, UPMCIrina Kourkova, UPMC

 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 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. AMS 2000 subject classifications: Primary 60J65, 05A15, 37L40; secondary 60K25, 30F10, 30D05Keywords: 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, Stochastic Systems, 7, (2017), 32-94 (electronic). DOI: 10.1214/16-SSY218.

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