Piecewise Deterministic Markov Processes (PDMP's) have been introduced in the literature by M.H.A. Davis as a general class of stochastics models. PDMP's are a family of Markov processes involving deterministic motion punctuated by random jumps. In a rst part, PDMP's are used to compute probabilities of top events for a case-study of dynamic reliability (the heated tank system) with two di erent methods : the rst one is based on the resolution of the di erential system giving the physical evolution of the tank and the second uses the computation of the functional of a PDMP by a system of integro-di erential equations. In the second part, we propose a numerical method to approximate the value function for the optimal stopping problem of a PDMP. Our approach is based on quantization of the postjump location and inter-arrival time of the Markov chain naturally embedded in the PDMP, and path-adapted time discretization grids. It allows us to derive bounds for the convergence rate of the algorithm and to provide a computable epsilon-optimal stopping time.
contribution à l'étude des processus Markoviens déterministes par morceaux. Etude d'un cas-test de la sûreté de fonctionnement et Problème d'arrêt optimal à horizon aléatoire