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A boundary property of semimartingale reflecting Brownian motions
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  • Published: March 1988

A boundary property of semimartingale reflecting Brownian motions

  • M. I. Reiman1 &
  • R. J. Williams2 

Probability Theory and Related Fields volume 77, pages 87–97 (1988)Cite this article

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  • 76 Citations

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Summary

We consider a class of reflecting Brownian motions on the non-negative orthant inR K. In the interior of the orthant, such a process behaves like Brownian motion with a constant covariance matrix and drift vector. At each of the (K-1)-dimensional faces that form the boundary of the orthant, the process reflects instantaneously in a direction that is constant over the face. We give a necessary condition for the process to have a certain semimartingale decomposition, and then show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces. This boundary property plays an essential role in the derivation (performed in a separate work) of an analytical characterization of the stationary distributions of such semimartingale reflecting Brownian motions.

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Author information

Authors and Affiliations

  1. AT&T Bell Laboratories, 07974, Murray Hill, NJ, USA

    M. I. Reiman

  2. University of California at San Diego, 92093, La Jolla, CA, USA

    R. J. Williams

Authors
  1. M. I. Reiman
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  2. R. J. Williams
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Additional information

Research performed in part while the second author was visiting the Institute for Mathematics and Its Applications with support provided by the National Science Foundation and the Air Force Office of Scientific Research. R.J. William's research was also supported in part by NSF Grant DMS 8319562.

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Cite this article

Reiman, M.I., Williams, R.J. A boundary property of semimartingale reflecting Brownian motions. Probab. Th. Rel. Fields 77, 87–97 (1988). https://doi.org/10.1007/BF01848132

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  • Received: 18 August 1986

  • Revised: 28 October 1987

  • Issue Date: March 1988

  • DOI: https://doi.org/10.1007/BF01848132

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Keywords

  • Covariance
  • Covariance Matrix
  • Stochastic Process
  • Brownian Motion
  • Essential Role
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