Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T21:42:28.475Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Asmussen, S. (1989). Risk theory in a Markovian environment. Scand. Actuarial J. 1989, No. 2, 69–100.Google Scholar
Asmussen, S. (2000). Ruin Probabilities. World Scientific, London.Google Scholar
Asmussen, S., and O'Cinneide, C. A. (1998). On the tail of the waiting time in a Markov-modulated M/G/1 queue. Tech. Rep. 18, Department of Mathematical Statistics, Lund University, Sweden.Google Scholar
Asmussen, S., and Rubinstein, R. Y. (1995). Steady state rare events simulation in queueing models and its complexity properties. In Advances in Queueing: Theory, Methods and Open Problems, ed. Dshalalow, J. H., CRC Press, Boca Raton, FL, pp. 429–461.Google Scholar
Dembo, A., and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.Google Scholar
Ellis, R. S. (1985). Entropy, Large Deviations and Statistical Mechanics. Springer, New York.Google Scholar
Lehtonen, T., and Nyrhinen, H. (1992). On asymptotically efficient simulation of ruin probabilities in a Markovian environment. Scand. Actuarial J. 1992, No. 1, 60–75.Google Scholar
Millet, A., Nualart, D., and Sanz, M. (1991). Composition of large deviations principles and applications. In Stochastic Analysis, eds Mayer-Wolf, E. et al., Academic Press, Boston, MA, pp. 383–395.Google Scholar