Some aspects of optimal quantization and applications to finance.

Summary This thesis is devoted to the study of optimal quantization and its applications. We deal with theoretical, algorithmic and numerical aspects. It consists of five chapters. In the first part, we study the links between variance reduction by stratification and quadratic optimal quantization. In the case where the random variable considered is a Gaussian process, a simulation scheme of linear complexity is developed for the conditional distribution at one stratum of the process in question. The second chapter is devoted to the numerical evaluation of the Karhunen-Loève basis of a Gaussian process by the Nyström method. In the third part, we propose a new approach to the quantization of EDS solutions, whose convergence we study. These results lead to a new cubature scheme for the solutions of stochastic differential equations, which is developed in the fourth chapter, and which we test on option pricing problems. In the fifth chapter, we present a new fast nearest neighbor tree search algorithm, based on the quantization of the empirical law of the considered point cloud.