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

Sandro Franceschi (1, 2) , Irina Kourkova (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Brownian motion in R 2 + with covariance matrix Σ and drift μ 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 (Σ, μ, 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.

Mots clés: Reflected Brownian motion in the quarter plane; Stationary distribution; Asymptotic analysis; Riemann surface; Saddle-point method; Laplace transform

Domaines: Probabilités [math.PR]

Sandro Franceschi, Irina Kourkova. Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach. Stochastic Systems, 2017, Volume 7, Number 1 (2017), 32-94., ⟨10.1214/16-SSY218⟩. ⟨hal-01295562v3⟩