EDSR: discretization analysis and resolution by adaptive Monte Carlo methods: domain perturbation for American options.

Summary Two different topics of numerical probabilities and their financial applications are addressed in my thesis: one deals with the approximation and simulation of backward-looking stochastic differential equations (SRDEs), the other is related to American options and approaches them from the point of view of domain optimization and frontier perturbations. The first part of my thesis revisits the issue of convergence analysis in the time discretization of Markovian (Y,Z) RDSRs into a dynamic programming equation of n time steps. We establish a first-order limited expansion of the error on (Y,Z): precisely, the trajectory error on X is fully transferred to the EDSR and thus show that if X is accurately approximated or simulated, better speeds are possible (in 1/n). The second part of my thesis focuses on the resolution of EDSRs via the Picard process and sequential Monte Carlo methods. We have shown that the convergence of our algorithm takes place at geometric speed and with an accuracy independent at the 1st order of the number of simulations. The last part of my thesis gathers first results on the valuation of American options by optimization of the exercise frontier. The keystone of this type of approach is the ability to evaluate a gradient with respect to the frontier. Continuous time has been treated by Costantini et al (2006) and this thesis covers the discrete case of Bermuda options.