Stochastic algorithms for risk management and indexing of media databases.

Summary This thesis deals with various control and optimization problems for which only approximate solutions exist to date. On the one hand, we are interested in techniques to reduce or eliminate approximations in order to obtain more precise or even exact solutions. On the other hand, we develop new approximation methods to deal more quickly with larger scale problems. We study numerical methods for simulating stochastic differential equations and for improving expectation calculations. We implement quantization-type techniques for the construction of control variables and the stochastic gradient method for solving stochastic control problems. We are also interested in clustering methods related to quantization, as well as in information compression by neural networks. The problems studied are not only motivated by financial issues, such as stochastic control for option hedging in incomplete markets, but also by the processing of large media databases commonly referred to as Big data in Chapter 5. Theoretically, we propose different majorizations of the convergence of numerical methods on the one hand for the search of an optimal hedging strategy in incomplete market in chapter 3, on the other hand for the extension of the Beskos-Roberts technique of differential equation simulation in chapter 4. We present an original use of the Karhunen-Loève decomposition for a variance reduction of the expectation estimator in chapter 2.