Lyapunov functions for semimartingale reflecting Brownian motions
type: Item
instance of: scholarly article
title: Lyapunov functions for semimartingale reflecting Brownian motions
zbMATH Open document ID: 0808.60068
author: Paul Dupuis
author: R. J. Williams
published in: The Annals of Probability
publication date: 1995-03-12T00:00:00Z
review text: Consider a nonnegative orthant \(S\) in \(R^ d\) in which a Brownian motion with constant drift \(r^ 0\) and covariance \(\Delta\) evolves. When it hits the \(i\)-th face of \(S\), it is reflected with direction \(r^ i\). Such a ``semimartingale reflecting Brownian motion'' (SRBM) associated with \((S,r^ 0, r^ 1, \dots, r^ d, \Delta )\) models certain problems in queueing networks under heavy traffic. The authors show that a sufficient condition for the SRBM to be positive recurrent is that all solutions of a related deterministic Skorokhod problem are attracted to the origin, i.e. the paths stay arbitrarily near zero after a time \(T\). The method of proving ergodicity uses the construction of a smooth Lyapunov function for the SRBM.
reviewed by: Michael Kohlmann
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 681378
zbMATH Keywords: semimartingale reflecting Brownian motion
zbMATH Keywords: recurrene
zbMATH Keywords: Skorokhod problem
zbMATH Keywords: Lyapunov function
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988725
OpenAlex ID: W2053963035
DOI: 10.1214/AOP/1176988725
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description: scientific article; zbMATH DE number 681378