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Continuous-time monotone stochastic recursions and duality

Part of: Special processes Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Karl Sigman  and
Reade Ryan
Karl Sigman*
Affiliation:
Columbia University
Reade Ryan*
Affiliation:
UCLA
*
∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, Mudd Bldg, MC: 4704, 500 West 120th Street, New York, NY 10027, USA.
∗∗ Postal address: The Anderson School at UCLA, 110 Westwood Plaza, Los Angeles, CA 90095-1481, USA. Email address: reade.ryan@anderson.ucla.edu
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Abstract

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

Keywords

Loynes' lemmajump-diffusion processreflected processrisk processruin probabilitySiegmund dualitystationary distributionstationary incrementsstorage processtwo-barrier reflection

MSC classification

Primary: 60G10: Stationary processes 60J60: Diffusion processes 60J75: Jump processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 426 - 445
Copyright
Copyright © Applied Probability Trust 2000 

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