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Boundary behavior and product-form stationary distributions of jump diffusions in the orthant with state-dependent reflections

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Francisco J. Piera ,
Ravi R. Mazumdar  and
Fabrice M. Guillemin
Francisco J. Piera*
Affiliation:
University of Chile
Ravi R. Mazumdar*
Affiliation:
University of Waterloo
Fabrice M. Guillemin*
Affiliation:
France Telecom R&D
*
∗ Postal address: Department of EE, University of Chile, Av. Tupper 2007, Santiago, 8370451, Chile. Email address: [email protected]
∗∗ Postal address: Department of ECE, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Email address: [email protected]
∗∗∗ Postal address: France Telecom R&D, 2 Avenue Pierre Marzin, 22300 Lannion, France. Email address: [email protected]
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Abstract

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In this paper we consider reflected diffusions with positive and negative jumps, constrained to lie in the nonnegative orthant of ℝn. We allow for the drift and diffusion coefficients, as well as for the directions of reflection, to be random fields over time and space. We provide a boundary behavior characterization, generalizing known results in the nonrandom coefficients and constant directions of the reflection case. In particular, the regulator processes are related to semimartingale local times at the boundaries, and they are shown not to charge the times the process expends at the intersection of boundary faces. Using the boundary results, we extend the conditions for product-form distributions in the stationary regime to the case when the drift and diffusion coefficients, as well as the directions of reflection, are random fields over space.

Keywords

Diffusionjumpsreflection mapregulator processlocal timesemimartingalestationary distributionproduct form

MSC classification

Primary: 60J60: Diffusion processes 60J75: Jump processes
Secondary: 60J50: Boundary theory 60J55: Local time and additive functionals 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 529 - 547
Copyright
Copyright © Applied Probability Trust 2008 

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