Mc Kean-Vlasov Stochastic Differential Equations and Their Optimal Control

We consider Mc Kean-Vlasov stochastic differential equations (SDEs), which are SDEs where the drift and diffusion coefficients depend not only on the state of the unknown process but also on its probability distribution. These SDEs called also mean- field SDEs were first studied in statistical physics and represent in some sense the average behavior of an infinite number of particles. Recently there has been a renewed interest for this kind of equations in the context of mean-field game theory. This theory was invented by P.L. Lions and J.M. Lasry in 2006, to solve the problem of existence of an approximate Nash equilibrium for differential games, with a large number of players. When the number of players tends to infinity, the equations defining the evolution of the particles could be replaced by a single equation, called the McKean-Vlasov equation. Since the pioneering papers by P.L. Lions and J.M. Lasry, mean-field games and mean-field control theory has raised a lot of interest, motivated by applications to various fields such as game theory, mathematical finance, communications networks and management of oil resources. The typical example is the continuous-time Markowitz mean-variance portfolio selection model in finance, where one should minimize an objective function involving a quadratic function of the expectation, due to the variance term. In this project, we plan to study questions of stability with respect to initial data, coefficients 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, will be investigated. In control theory, our attention will be focused on the maximum principle for systems driven by Mc Kean-Vlasov SDEs controlled by both regular as well as singular controls