Stochastic control by quantization methods and applications to finance.

Summary This thesis contains three parts that can be read independently. In the first part, we study the resolution of stochastic control problems by quantization methods. The quantization consists in finding the best approximation of continuous probability distribution by a discrete probability law with a number N of points supporting this distribution. We explicit a framework of “generic” dynamic programming which permits to resolve many stochastic control problems, such as optimal stopping time problems, maximization of utility, backward stochastic differential equations (BSDE), filter problems… In this context, we give three discretization schemes in space associated to the quantization of a Markov chain. In the second part, we present a numerical scheme for doubly reflected BSDEs. We consider a general framework which contains jumps and path-dependent progressive processes. We use a discrete time Euler-type approximation scheme. We prove the convergence of this scheme for BSDE when the time step number n tends to infinity. We also give the convergence speed for game options. In the third part, we focus on the replication of derivatives on realized variance. We suggest a robust hedging to the volatility model with dynamic positions on European options. Then, we extend this methodology to fund options and to jump process.