Some results on the McKean–Vlasov optimal control and mean field games : Limit theorems, dynamic programming principle and numerical approximations.

Summary This thesis is formulated in three parts with eight chapters and presents a research theme dealing with controlled processes/particles/interacting agents.In the first part of the thesis, we focus our attention on the study of interacting controlled processes representing a cooperative equilibrium, also known as Pareto equilibrium. A cooperative equilibrium can be seen as a situation where there is no way to improve the preference criterion of one agent without lowering the preference criterion of at least one other agent. It is now well known that this type of optimization problem is related, when the number of agents goes to infinity, to McKean-Vlasov optimal control. In the first three chapters of this thesis, we provide a precise mathematical answer to the link between these two optimization problems in different frameworks improving the existing literature, in particular by taking into account the control law while allowing a common noise situation.After studying the behavior of cooperative equilibria, we conclude the first part where we spend time in the analysis of the limit problem i.e. McKean-Vlasov optimal control, through the establishment of the dynamic programming principle (DPP) for this stochastic control problem.The second part of this thesis is devoted to the study of interacting controlled processes now representing a Nash equilibrium, also known as competitive equilibrium. A Nash equilibrium situation in a game is one in which no one has anything to gain by unilaterally leaving his own position. Since the pioneering work of Larsy - Lions and Huang - Malhamé - Caines, the behavior of Nash equilibria when the number of agents reaches infinity has been intensively studied and the associated limit game is known as Mean Field Games (MFG). In this second part, we first analyze the convergence of competitive equilibria to MFGs in a framework with the control law and with volatility control, then, the question of the existence of the MFG equilibrium in this context is studied.Finally, the last part, which consists of only one chapter, is devoted to some numerical methods for solving the limit problem i.e. McKean - Vlasov optimal control. Inspired by the proof of convergence of the cooperative equilibrium, we give a numerical algorithm to solve the McKean-Vlasov optimal control problem and prove its convergence. Then, we implement our algorithm using neural networks and test its efficiency on some application examples, namely mean-variance portfolio selection, the interbank systemic risk model and optimal liquidation with market impact.