Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant
type: Item
instance of: scholarly article
title: Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant
zbMATH Open document ID: 0794.60079
DOI: 10.1007/BF01292674
publication date: 1994-08-15T00:00:00Z
review text: This work is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in the nonnegative orthant of \(\mathbb{R}^ d\). Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the \((d-1)\)-dimensional faces that form the boundary of the orthant, the bounded variation part of the process increases in a given direction (constant for any particular face) so as to confine the process to the orthant. For historical reasons, this ``pushing'' at the boundary is called instantaneous reflection. \textit{M. I. Reiman} and \textit{R. J. Williams} [ibid. 77, No. 1, 87-97 (1988; Zbl 0617.60081)] proved that a necessary condition for the existence of such a semimartingale reflecting Brownian motion (SRBM) is that the reflection matrix formed by the directions of reflection be completely-\({\mathcal S}\). We prove that this condition is sufficient for the existence of an SRBM and that the SRBM is unique in law. It follows from the uniqueness that an SRBM defines a strong Markov process. Our results have potential application to the study of diffusions arising as approximations to multiclass queueing networks.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 569079
zbMATH Keywords: reflecting Brownian motions
zbMATH Keywords: semimartingale decomposition
zbMATH Keywords: strong Markov process
zbMATH Keywords: multiclass queueing networks
author: Lee M. Taylor
author: R. J. Williams
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 569079