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A Brownian motion with two reflecting barriers and Markov-modulated speed

Part of: Markov processes Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella  and
Wolfgang Stadje
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Wolfgang Stadje*
Affiliation:
University of Osnabrück
*
∗ Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: [email protected]
∗∗ Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: [email protected]
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Abstract

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.

Keywords

Brownian motionMarkov modulatedstate-dependent speedreflected Brownian motiontwo-sided reflectiondouble barriermultidimensional martingale

MSC classification

Primary: 60K37: Processes in random environments
Secondary: 60J65: Brownian motion 90B05: Inventory, storage, reservoirs
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1237 - 1242
Copyright
Copyright © Applied Probability Trust 2004 

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Footnotes

This research was supported by the Volkswagen foundation.

References

Asmussen, S., and Kella, O. (2000). A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Prob. 32, 376–393.Google Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar