Acceleration of the Monte Carlo method for diffusion processes and applications in Finance.

Summary In this thesis, we focus on the combination of variance reduction and complexity reduction methods of the Monte Carlo method. In a first part of this thesis, we consider a continuous diffusion model for which we build an adaptive algorithm by applying importance sampling to the Romberg Statistical method. We prove a Lindeberg Feller type central limit theorem for this algorithm. In this same framework and in the same spirit, we apply importance sampling to the Multilevel Monte Carlo method and we also prove a central theorem for the obtained adaptive algorithm. In the second part of this thesis, we develop the same type of algorithm for a non-continuous model, namely the Lévy processes. Similarly, we prove a central limit theorem of the Lindeberg Feller type. Numerical illustrations have been carried out for the different algorithms obtained in the two frameworks with jumps and without jumps.