Portfolio optimization in financial markets with partial information.

Summary This thesis deals - in three essays - with portfolio choice problems in a situation of partial information, a theme that we present in a short introduction. The essays developed each address a particularity of this problem. The first essay (co-written with M. Jeanblanc and V. Lacoste) deals with the question of choosing the optimal strategy for a terminal utility maximization problem when the evolution of prices is modeled by an Itô-Lévy process whose trend and intensity of jumps are not observed. The approach consists in rewriting the initial problem as a reduced problem in the filtration generated by the prices. This requires the derivation of the nonlinear filtering equations, which we develop for a Lévy process. The problem is then solved using dynamic programming by the Bellman and HJB equations. The second essay addresses in a Gaussian framework the question of the cost of uncertainty, which we define as the difference between the optimal strategies (or maximum wealth) of a perfectly informed agent and a partially informed agent. The properties of this cost of information are studied in the context of the three standard forms of utility functions and numerical examples are presented. Finally, the third essay addresses the issue of portfolio choice when market price information is only available at discrete and random dates. This amounts to assuming that price dynamics follow a marked process. In this framework, we develop the filtering equations and rewrite the initial problem in its reduced form in discrete price filtration. The optimal strategies are then computed using the Malliavin calculus for random measures and an extension of the Clark-Ocone-Haussman formula is presented for this purpose.