Journal of Applied Mathematics and Stochastic Analysis 
Volume 2006 (2006), Article ID 92156, 18 pages
doi:10.1155/JAMSA/2006/92156

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

Abstract

In dynamic reliability, the evolution of a system is described by a piecewise deterministic Markov process (It,Xt)t≥0 with state-space E×ℝd, where E is finite. The main result of the present paper is the characterization of the marginal distribution of the Markov process (It,Xt)t≥0 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.