Recursive approximation of the stationary regime of a stochastic differential equation with jumps.

Summary This thesis is mainly devoted to the construction and study of computer-implementable methods for approximating the stationary regime of a multidimensional ergodic process, solution of an EDS directed by a Lévy process. Based on an approach developed by Lamberton&Pagès and Lemaire in the framework of Brownian diffusions, our methods based on "exact" or "approximated" decreasing step Euler schemes allow to efficiently simulate the invariant probability but also the global law of such a process in stationary regime. This work has various theoretical and practical applications, some of which are developed here (TCL p. S. For stable laws, limit theorem for extreme values, option pricing for stationary stochastic volatility models. . ).