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On product-form stationary distributions for reflected diffusions with jumps in the positive orthant

Part of: Markov processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Francisco J. Piera ,
Ravi R. Mazumdar  and
Fabrice M. Guillemin
Francisco J. Piera*
Affiliation:
Purdue University
Ravi R. Mazumdar*
Affiliation:
University of Waterloo and Purdue University
Fabrice M. Guillemin*
Affiliation:
France Télécom
*
∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA. Email address: fpieraug@purdue.edu
∗∗ Postal address: Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Email address: mazum@ece.uwaterloo.ca
∗∗∗ Postal address: France Télécom R&D, 2 Avenue Pierre Marzin, 22300 Lannion, France. Email address: fabrice.guillemin@francetelecom.com
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Abstract

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In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R+n that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

Keywords

Diffusionjumpreflection maplocal timesemimartingalestationary distributionproduct form

MSC classification

Primary: 60J50: Boundary theory 60J55: Local time and additive functionals
Secondary: 60J60: Diffusion processes 60J75: Jump processes 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 212 - 228
Copyright
Copyright © Applied Probability Trust 2005 

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