International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2006 / Article

Open Access

Volume 2006 |Article ID 092156 | https://doi.org/10.1155/JAMSA/2006/92156

Christiane Cocozza-Thivent, Robert Eymard, Sophie Mercier, Michel Roussignol, "Characterization of the marginal distributions of Markov processes used in dynamic reliability", International Journal of Stochastic Analysis, vol. 2006, Article ID 092156, 18 pages, 2006. https://doi.org/10.1155/JAMSA/2006/92156

Characterization of the marginal distributions of Markov processes used in dynamic reliability

Received03 May 2004
Revised15 Feb 2005
Accepted22 Feb 2005
Published05 Mar 2006

Abstract

In dynamic reliability, the evolution of a system is described by a piecewise deterministic Markov process (It,Xt)t0 with state-space E×d, where E is finite. The main result of the present paper is the characterization of the marginal distribution of the Markov process (It,Xt)t0 at time t, as the unique solution of a set of explicit integro-differential equations, which can be seen as a weak form of the Chapman-Kolmogorov equation. Uniqueness is the difficult part of the result.

References

  1. H. Amann, Ordinary Differential Equations, vol. 13 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, 1990. View at: Zentralblatt MATH | MathSciNet
  2. K. Borovkov and A. Novikov, “On a piece-wise deterministic Markov process model,” Statistics & Probability Letters, vol. 53, no. 4, pp. 421–428, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. C. Cocozza-Thivent, R. Eymard, and S. Mercier, “A Numerical scheme to solve integro-differential equations in the dynamic reliability field,” in PSAM 7/ESREL '04, Special Technical Session: “Dynamic reliability: Methods and applications”, pp. 675–680, Springer, Berlin, 2004. View at: Google Scholar
  4. C. Cocozza-Thivent, R. Eymard, and S. Mercier, “A numerical scheme for solving integro-differential equations from the dynamic reliability field,” to appear in IMA Journal of Numerical Analysis. View at: Google Scholar
  5. O. L. V. Costa and F. Dufour, “On the poisson equation for piecewise-deterministic Markov processes,” SIAM Journal on Control and Optimization, vol. 42, no. 3, pp. 985–1001, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. O. L. V. Costa, C. A. B. Raymundo, and F. Dufour, “Optimal stopping with continuous control of piecewise deterministic Markov processes,” Stochastics and Stochastic Reports, vol. 70, no. 1-2, pp. 41–73, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  7. M. H. A. Davis, “Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models,” Journal of the Royal Statistical Society. Series B. Methodological, vol. 46, no. 3, pp. 353–388, 1984. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. M. H. A. Davis, Markov Models and Optimization, vol. 49 of Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1993. View at: Zentralblatt MATH | MathSciNet
  9. F. Dufour and O. L. V. Costa, “Stability of piecewise-deterministic Markov processes,” SIAM Journal on Control and Optimization, vol. 37, no. 5, pp. 1483–1502, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, New York, 1986. View at: Zentralblatt MATH | MathSciNet
  11. R. A. Holley and D. W. Stroock, “A martingale approach to infinite systems of interacting processes,” The Annals of Probability, vol. 4, no. 2, pp. 195–228, 1976. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. P. E. Labeau, Méthodes semi-analytiques et outils de simulation en dynamique probabiliste, M.S. thesis, Faculté des Sciences Appliquées, Service de Métrologie Nucléaire, ULB, Bruxelles, 1996.
  13. T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels, Stochastic Processes for Insurance and Finance, Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester, 1999. View at: Zentralblatt MATH | MathSciNet

Copyright © 2006 Christiane Cocozza-Thivent et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Order printed copiesOrder
Views86
Downloads427
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.