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Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Sean P. Meyn  and
R. L. Tweedie
Sean P. Meyn*
Affiliation:
University of Illinois
R. L. Tweedie*
Affiliation:
Colorado State University
*
∗ Postal address: University of Illinois, Coordinated Science Laboratory, 1101 W. Springfield Ave., Urbana, IL 61801, USA.
∗∗ Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA.
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Abstract

In Part I we developed stability concepts for discrete chains, together with Foster–Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes. In this paper we develop criteria for these forms of stability for continuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator.

Such test function criteria are found for non-explosivity, non-evanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula.

We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, and bounding constants for such convergence in the exponential ergodic case.

We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes.

Keywords

FOSTER'S CRITERIONIRREDUCIBLE MARKOV PROCESSESSTOCHASTIC LYAPUNOVFUNCTIONSERGODICITYEXPONENTIAL ERGODICITYRECURRENCESTORAGE MODELSRISK MODELSQUEUESHYPOELLIPTIC DIFFUSION

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 518 - 548
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

This work was commenced at Bond University, and developed there, at the Australian National University, the University of Illinois, and Colorado State University.

Work supported in part by NSF initiation grant #ECS 8910088.

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