Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution

J G Dai,Masakiyo Miyazawa

doi:10.1214/10-ssy022

Abstract

Open AccessOpen Access licenseAboutSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail Go to SectionOpen AccessOpen Access license HomeStochastic SystemsVol. 1, No. 1 Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary DistributionJ. G. Dai, Masakiyo MiyazawaJ. G. Dai, Masakiyo MiyazawaPublished Online:15 Jul 2011https://doi.org/10.1287/10-SSY022AbstractWe consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form bxκ exp(−αx) as x goes to infinity, where α and b are positive constants and κ takes one of the values −3/2, −1/2, 0, or 1; both the decay rate α and the power κ can be computed explicitly from the given direction and the SRBM data.A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of the two-dimensional moment generating function. For a given direction c, the line in this direction intersects the boundary of the convergence domain at one point, and that point uniquely determines the decay rate α. The one-dimensional moment generating function of the marginal distribution along direction c has a singularity at α. Using analytic extension in complex analysis, we characterize the precise nature of the singularity there. Using that characterization and complex inversion techniques, we obtain the exact asymptotic of the marginal tail distribution. Previous Back to Top FiguresReferencesRelatedInformationCited BySplitting Algorithms for Rare Events of Semimartingale Reflecting Brownian MotionsKevin Leder, Xin Liu, Zicheng Wang2 September 2021 | Stochastic Systems, Vol. 11, No. 4Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic ApproachSandro Franceschi, Irina Kourkova8 May 2017 | Stochastic Systems, Vol. 7, No. 1Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the OctantZiyu Liang, John J. Hasenbein19 August 2013 | Stochastic Systems, Vol. 3, No. 1Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected ProcessesMasakiyo Miyazawa, Bert Zwart7 May 2012 | Stochastic Systems, Vol. 2, No. 1 Volume 1, Issue 1June 2011Pages 1-208 Article Information Metrics Downloaded 67 times in the past 12 months Information Received:November 01, 2010Published Online:July 15, 2011 Copyright © 2011, The author(s)Cite asJ. G. Dai, Masakiyo Miyazawa (2011) Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary Distribution. Stochastic Systems 1(1):146-208. https://doi.org/10.1287/10-SSY022 KeywordsDiffusion processheavy trafficqueueing networktail behaviorlarge deviationsPDF download

Highlights

This paper is concerned with the asymptotic tail behavior of the stationary distributions of two-dimensional semimartingale reflecting Brownian motions (SRBMs)
For many cases we prove that the density pc(x) has an exact asymptotic, which implies that the exact asymptotic for the tail distribution qc(x) = P( c, Z(∞) ≥ x), x ≥ 0 by Lemma D.5
They are the non-singular assumption on the covariance matrix Σ, the large deviations rate function, the fine exact asymptotics, and an extension to SRBMs in more than two dimensions

Summary

Introduction

This paper is concerned with the asymptotic tail behavior of the stationary distributions of two-dimensional semimartingale reflecting Brownian motions (SRBMs). The convergence domain (1) of the two-dimensional moment generating function φ(θ) defined in (1.9) is of primary importance in determining the asymptotic tail of P{ c, Z(∞) ≥ u} as u → ∞. The following key relationship among moment generating functions holds: (2.3)

Results

Conclusion