[Paper Review] Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures

J. G. Dai, Masakiyo Miyazawa|arXiv (Cornell University)|Oct 9, 2011

Stochastic processes and statistical mechanics9 references4 citations

TL;DR

This paper provides a geometric characterization of the stationary distribution of a two-dimensional semimartingale reflecting Brownian motion (SRBM) using an ellipse and two boundary lines derived from the SRBM's covariance, drift, and reflection matrices. It establishes a geometric interpretation of the variational problem's value function, derives a condition for product-form stationary distributions, and proves exact tail asymptotics for boundary measures by refining asymptotic inversion lemmas for moment generating functions.

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) cap 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.

Motivation & Objective

• To provide a geometric interpretation of the variational problem's value function in two-dimensional SRBM.
• To derive a geometric condition for the existence of a product-form stationary distribution in 2D SRBM.
• To establish exact tail asymptotics for boundary measures associated with the stationary distribution of 2D SRBM.
• To refine asymptotic inversion lemmas for moment generating functions to enable exact tail analysis.

Proposed method

• The SRBM data is represented geometrically via an ellipse (from drift and covariance) and two lines (from reflection matrix columns), forming three key geometric objects.
• The value function of the variational problem is expressed as the minimum of two convex optimization problems over domains D^(1) and D^(2), which are defined by the geometric objects.
• The exit velocities from boundaries are characterized as normal vectors to the ellipse at symmetry points of the reflection directions.
• A refined version of the asymptotic inversion lemma (Lemma B.1) is used to infer the exact tail behavior of boundary measures from singularities of their moment generating functions.
• The proof of tail asymptotics relies on analyzing the analytic structure of the moment generating function near its dominant singularity.
• The product-form condition is derived geometrically by analyzing the alignment of the reflection directions and the ellipse’s principal axes.

Experimental results

Research questions

• RQ1 How can the value function of the variational problem for a 2D SRBM be interpreted geometrically in terms of the ellipse and boundary lines?
• RQ2 Under what geometric condition does a 2D SRBM admit a product-form stationary distribution?
• RQ3 What is the exact tail asymptotic behavior of the boundary measures associated with the stationary distribution of a 2D SRBM?
• RQ4 How can asymptotic inversion lemmas be sharpened to derive exact tail asymptotics from moment generating function singularities?

Key findings

• The value function of the variational problem at any point in the nonnegative quadrant equals the minimum of two convex optimizations over domains D^(1) and D^(2), which are geometrically defined by the ellipse and the two boundary lines.
• The exit velocity from boundary F_i is given by the normal vector to the ellipse at the symmetry point of the reflection direction R^i, providing a geometric characterization of optimal paths.
• A necessary and sufficient geometric condition for a product-form stationary distribution is derived, involving the alignment of the reflection directions with the ellipse’s principal axes.
• The boundary measures exhibit exact exponential tail asymptotics of the form f(x) ∼ c x^{ℓ−1} e^{−αx} as x→∞, determined by the dominant singularity of the moment generating function.
• The asymptotic inversion lemma (Lemma B.1) is refined to ensure that the tail behavior of the boundary measure is precisely determined by the order and location of the dominant pole of the moment generating function.
• The exact tail asymptotics are established by proving uniform convergence and analyticity conditions on the moment generating function, enabling precise asymptotic inference.