Numerical methods for piecewise deterministic Markovian processes.

Summary Piecewise Deterministic Markovian Processes (PDMP) were introduced in the literature by M.H.A. Davis as a general class of non-diffusive stochastic models. PMDMs are hybrid processes characterized by deterministic trajectories interspersed with random jumps. In this thesis, we develop numerical methods adapted to PMDMs based on the quantization of a Markov chain underlying the PMDM. We successively address three problems: the approximation of function expectations of a PMDM, the approximation of the moments and distribution of an output time and the partially observed optimal stopping problem. In this last part, we also address the issue of filtering a PMDM and establish the dynamic programming equation of the optimal stopping problem. We prove the convergence of all our methods (with bounds on the convergence speed) and illustrate them with numerical examples.