Markov chains approximation of jump-diffusion stochastic master equations

Quantum trajectories are solutions of stochastic differential equations obtained when describing the random phenomena associated to quantum continuous measurement of open quantum system. These equations, also called Belavkin equations or Stochastic Master equations, are usually of two different types: diffusive and of Poisson-type. In this article, we consider more advanced models in which jump diffusion equations appear. These equations are obtained as a continuous time limit of martingale problems associated to classical Markov chains which describe quantum trajectories in a discrete time model. The results of this article goes much beyond those of [Ann. Probab. 36 (2008) 2332-2353] and [Existence, uniqueness and approximation for stochastic Schrodinger equation: The Poisson case (2007)]. The probabilistic techniques, used here, are completely different in order to merge these two radically different situations: diffusion and Poisson-type quantum trajectories.