Dynamic error control for simulation and estimation of diffusion processes.

Summary The asymptotic properties of Monte-Carlo type algorithms and of the usual functionals of ergodic diffusion processes are characterized using central limit theorems. The purpose of this thesis is to present results refining these theorems in four different settings. The first part of this work concerns the simulation of diffusion processes. The first chapter is devoted to the presentation of a method for adaptively controlling the variance during a Monte-Carlo simulation. Applications are given in finance. The second chapter proposes an estimator of the asymptotic variance of ergodic simulations. Its construction is based on results of the almost sure central limit theorem. Variance reduction techniques are proposed in this framework. The second part concerns the statistics of processes. The first chapter deals with ergodic diffusion processes. For different functionals of these processes, we prove Edgeworth developments specifying the speed of convergence of the central limit theorem. Applications in statistics are proposed, and in particular an opening towards the bootstrap. The second chapter proposes a theoretical framework for the parametric estimation of diffusion processes generalizing the asymmetric Brownian motion.