Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 92156, 18 pages

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

Christiane Cocozza-Thivent, Robert Eymard, Sophie Mercier, and Michel Roussignol

Laboratoire d'Analyse et de Mathématiques Appliquées \newline (CNRS UMR 8050), Université de Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, Marne-la-Vallée cedex 2 77454, France

Received 3 May 2004; Revised 15 February 2005; Accepted 22 February 2005

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.


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.