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Continuous-time Markov additive processes: Composition of large deviations principles and comparison between exponential rates of convergence

Part of: Limit theorems Special processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Claudio Macci
Claudio Macci*
Affiliation:
Università degli Studi di Torino
*
∗ Postal address: Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto 10, I-10123 Torino, Italy. Email address: macci@dm.unito.it
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Abstract

We consider a continuous-time Markov additive process (Jt,St) with (Jt) an irreducible Markov chain on E = {1,…,s}; it is known that (St/t) satisfies the large deviations principle as t → ∞. In this paper we present a variational formula H for the rate function κ∗ and, in some sense, we have a composition of two large deviations principles. Moreover, under suitable hypotheses, we can consider two other continuous-time Markov additive processes derived from (Jt,St): the averaged parameters model (Jt,St(A)) and the fluid model (Jt,St(F)). Then some results of convergence are presented and the variational formula H can be employed to show that, in some sense, the convergences for (Jt,St(A)) and (Jt,St(F)) are faster than the corresponding convergences for (Jt,St).

Keywords

Markov additive processeslarge deviationslevel crossing probabilitiesimportance sampling

MSC classification

Primary: 60F10: Large deviations 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 917 - 931
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

This work has been partially supported by Murst Project ‘Processi stocastici, calcolo stocastico ed applicazioni’.

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