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April, 1981 Reflected Brownian Motion on an Orthant
J. Michael Harrison, Martin I. Reiman
Ann. Probab. 9(2): 302-308 (April, 1981). DOI: 10.1214/aop/1176994471


We consider a $K$-dimensional diffusion process $Z$ whose state space is the nonnegative orthant. On the interior of the orthant, $Z$ behaves like a $K$-dimensional Brownian motion with arbitrary covariance matrix and drift vector. At each of the (`K-1) dimensional hyperplanes that form the boundary of the orthant, $Z$ reflects instantaneously in a direction that is constant over that hyperplane. There is no extant theory of multidimensional diffusion that applies to this process, because the boundary of its state space is not smooth. We adopt an approach that requires a restriction on the directions of reflection, but Reiman has shown that this restriction is met by all diffusions $Z$ arising as heavy traffic limits in open $K$-station queuing networks. Our process $Z$ is defined as a path-to-path mapping of $K$-dimensional Brownian motion. From this construction it follows that $Z$ is a continuous Markov process and a semimartingale. Using the latter property, we obtain a change of variable formula from which one can develop a complete analytical theory for the process $Z$.


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J. Michael Harrison. Martin I. Reiman. "Reflected Brownian Motion on an Orthant." Ann. Probab. 9 (2) 302 - 308, April, 1981.


Published: April, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60073
MathSciNet: MR606992
Digital Object Identifier: 10.1214/aop/1176994471

Primary: 60J50
Secondary: 60J55 , 60J60 , 60J65

Keywords: Brownian motion , Diffusion processes , heavy traffic , Ito's formula , Markov processes , queuing networks , reflecting barriers , Semimartingale

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 2 • April, 1981
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