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Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant
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  • Published: September 1993

Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant

  • L. M. Taylor1 &
  • R. J. Williams2 

Probability Theory and Related Fields volume 96, pages 283–317 (1993)Cite this article

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Summary

This work is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in the non-negative orthant of ℝd. 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. In 1988, Reiman and Williams 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-L. In this work we prove that 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 tomulti-class queueing networks.

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Author information

Authors and Affiliations

  1. Department of Mathematics, California State University, 95819-6051, Sacramento, CA, USA

    L. M. Taylor

  2. Department of Mathematics, University of California, 92093-0112, San Diego, CA, USA

    R. J. Williams

Authors
  1. L. M. Taylor
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  2. R. J. Williams
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Additional information

Research supported in part by NSF Grants DMS 8657483, 8722351 and 9023335, and a grant from AT&T Bell Labs. In addition, R.J. Williams was supported in part during the period of this research by an Alfred P. Sloan Research Fellowship

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Taylor, L.M., Williams, R.J. Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probab. Th. Rel. Fields 96, 283–317 (1993). https://doi.org/10.1007/BF01292674

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  • Received: 09 May 1991

  • Revised: 23 October 1992

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01292674

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Mathematics Subject Classification (1991)

  • 60J60
  • 60J65
  • 60G44
  • 60K25
  • 58G32
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