Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures

Abstract

We present three sets of results for the stationary distribution of a two-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative quadrant. The SRBM data can equivalently be specified by three geometric objects, an ellipse and two lines, in the two-dimensional Euclidean space. First, we revisit the variational problem (VP) associated with the SRBM. Building on Avram, Dai and Hasenbein (2001), we show that the value of the VP at a point in the quadrant is equal to the optimal value of a linear function over a convex domain. Depending on the location of the point, the convex domain is either D (1) or D (2) or D (1) ∩ D (2) , where each D (i) , i = 1, 2, can easily be described by the three geometric objects. Our results provide a geometric interpretation for the value function of the VP and allow one to see geometrically when one edge of the quadrant has influence on the optimal path traveling from the origin to a destination point. Second, we provide a geometric condition that characterizes the existence of a product form stationary distribution. Third, we establish exact tail asymptotics of two boundary measures that are associated with the stationary distribution; a key step in our proof is to sharpen two asymptotic inversion lemmas in Dai and Miyazawa (2011) that allow one to infer the exact tail asymptotic of a boundary measure from the singularity of its moment generating function.

Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

“…Let us note that the exponents in Theorem 4 are the same as in the large deviation rate function found in [7,Thm 3.2]. The same phenomenon is observed for discrete random walks, cf.…”

“…Two-dimensional semimartingale reflecting Brownian motion (SRBM) in the quarter plane received a lot of attention from the mathematical community. Problems such as SRBM existence [39,40], stationary distribution conditions [19,22], explicit forms of stationary distribution in special cases [7,8,19,23,30], large deviations [1,7,33,34] construction of Lyapunov functions [10], and queueing networks approximations [19,21,31,32,43] have been intensively studied in the literature. References cited above are non-exhaustive, see also [42] for a survey of some of these topics.…”

“…In [6], Dai et Myazawa obtain the following asymptotic result: for a given directional vector c ∈ R 2 + they find the function f c (x) such that where · | · is the inner product. In [7] they compute the exact asymptotics of two boundary stationary measures on the axes associated with Z(∞).…”

“…In the cases α = 0 and α = π/2, the tail asymptotics of the boundary measures ν 1 and ν 2 has been found in [7] and the constants have been specified in [18]. It would be also possible to find the asymptotics of π(r cos α, r sin α) where r → ∞ and α → 0 or α → π/2.…”

Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

“…Hence, ϕ(zc) is analytically extendable for Re z < α c + ε except at the simple pole at z = α c . To this analytic function, we apply Lemma 6.2 of [7], which is an adaptation of the asymptotic inversion due to Doetsch [8]. This yields (12).…”