Probabilistic numerical methods for solving some multidimensional nonlinear problems in finance.

Summary This thesis consists of 3 parts. The first part is devoted to over-replication problems in financial asset models with "uncertain volatility" (UV). The second part deals with probabilistic methods for solving some nonlinear multidimensional problems in finance, such as the pricing and hedging of American options on a basket of underlyings. In the third part, some of these methods are applied to the problems presented in the first part. The first part recalls the main results of the literature for stocks and extends them to interest rates by introducing an HJM framework with uncertain volatility. The second part is devoted to the acceleration of various probabilistic numerical methods for the solution of optimal stopping time problems. Three types of approaches are presented: regression methods (Longstaff-Schwartz, Tsitsiklis-Van Roy), Markov chain approximation methods (Broadie-Glasserman, Quantification) and Monte-Carlo Malliavin type methods. In the third part, we apply successively quantification methods and Monte-Carlo Malliavin type methods to the valuation and hedging of European options on a multidimensional underlying in a UV framework.