Variance reduction and discretization of stochastic differential equations. Almost sure limit theorems for left-handed quasi-continuous martingales.

Summary This Thesis is composed of two parts dealing respectively with the discretization of stochastic differential equations and the almost sure central limit theorem for martingales. The first part is composed of three chapters: The first chapter introduces the framework of the study and presents the results obtained. The second chapter is devoted to the study of a new convergence acceleration method, called the statistical Romberg method, for the computation of expectations of functions or functionals of a diffusion. This chapter is an extended version of an article to be published in the journal Annals of Applied Probability. The third chapter deals with the application of this method to density approximation by kernel methods. This chapter is based on a collaborative work with Arturo Kohatsu-Higa. The second part of the thesis is composed of two chapters: the first chapter presents the recent literature concerning the almost sure central limit theorem and its extensions. The second chapter, based on a collaborative work with Faouzi Chaâbane, extends various TLCPS results to quasi-continuous left-handed martingales.