Christiane Cocozza-Thivent

This book is aimed at researchers, graduate students and engineers who would like to be initiated to Piecewise Deterministic Markov Processes (PDMPs). A PDMP models a deterministic mechanism modified by jumps that occur at random times. The fields of applications are numerous : insurance and risk, biology, communication networks, dependability, supply management, etc. Indeed, the PDMPs studied so far are in fact deterministic functions of CSMPs (Completed Semi-Markov Processes), i.e. semi-Markov processes completed to become Markov processes. This remark leads to considerably broaden the definition of PDMPs and allows their properties to be deduced from those of CSMPs, which are easier to grasp. Stability is studied within a very general framework. In the other chapters, the results become more accurate as the assumptions become more precise. Generalized Chapman-Kolmogorov equations lead to numerical schemes. The last chapter is an opening on processes for which the deterministic flow of the PDMP is replaced with a Markov process. Marked point processes play a key role throughout this book.

This book is aimed at researchers, graduate students and engineers who would like to be initiated to Piecewise Deterministic Markov Processes (PDMPs). A PDMP models a deterministic mechanism modified by jumps that occur at random times. The fields of applications are numerous : insurance and risk, biology, communication networks, dependability, supply management, etc. Indeed, the PDMPs studied so far are in fact deterministic functions of CSMPs (Completed Semi-Markov Processes), i.e. semi-Markov processes completed to become Markov processes. This remark leads to considerably broaden the definition of PDMPs and allows their properties to be deduced from those of CSMPs, which are easier to grasp. Stability is studied within a very general framework. In the other chapters, the results become more accurate as the assumptions become more precise. Generalized Chapman-Kolmogorov equations lead to numerical schemes. The last chapter is an opening on processes for which the deterministic flow of the PDMP is replaced with a Markov process. Marked point processes play a key role throughout this book.

Ce livre, construit à partir de cours de DESS et de DEA, peut aussi intéresser des étudiants de maîtrise et des ingénieurs. Son fil directeur est la fiabilité et son but est de montrer ce que peut apporter à ce domaine l'étude des processus stochastiques. Chemin faisant, cela permet d'aborder, dans des cas simples, des techniques variées. Certaines parties sont d'inspiration très appliquée et peuvent être lues par toute personne ayant des connaissances de base en probabilités-statistiques. D'autres font appel à des concepts plus pointus et offrent des ouvertures sur la recherche. Le plus souvent un thème donne lieu à deux chapitres: le premier présente les outils mathématiques, le second les applications à la fiabilité.