Equations différentielles stochastiques de type McKean-Vlasov et leur contrôle optimal

Abstract

We consider Mc Kean-Vlasov stochastic di¤erential equations (SDEs), which are SDEs where the drift and di¤usion coe¢ cients depend not only on the state of the unknown process but also on its probability distribution. These SDEs called also mean- …eld SDEs were …rst studied in statistical physics and represent in some sense the average behavior of an in…nite number of particles. Recently there has been a renewed interest for this kind of equations in the context of mean-…eld game theory. Since the pioneering papers by P.L. Lions and J.M. Lasry, mean-…eld games and mean-…eld control theory has raised a lot of interest, motivated by applications to various …elds such as game theory, mathematical …nance, communications networks and management of oil resources. In this thesis, we studied questions of stability with respect to initial data, coe¢ cients and driving processes of Mc Kean-Vlasov equations. Generic properties for this type of SDEs, such as existence and uniqueness, stability with respect to parameters, have been investigated. In control theory, our attention were focused on existence, approximation of relaxed controls for controlled Mc Kean-Vlasov SDEs.