Topic in mean field games theory & applications in economics and quantitative finance.

Summary Mean Field Game (MFG) systems describe equilibrium configurations in differential games with infinitely many infinitesimal interacting agents. This thesis is articulated around three different contributions to the theory of Mean Field Games. The main purpose is to explore the power of this theory as a modeling tool in various fields, and to propose original approaches to deal with the underlying mathematical questions. The first chapter presents the key concepts and ideas that we use throughout the thesis: we introduce the MFG problem, and we briefly explain the asymptotic link with N-Player differential games when N → ∞. Next we present our main results and contributions, that are explained more in details in the subsequent chapters. In Chapter 2, we explore a Mean Field Game model with myopic agents. In contrast to the classical MFG models, we consider less rational agents which do not anticipate the evolution of the environment, but only observe the current state of the system, undergo changes and take actions accordingly. We analyze the resulting system of coupled PDEs and provide a rigorous derivation of that system from N-Player stochastic differential games models. Next, we show that our population of agents can self-organize and converge exponentially fast to the well-known ergodic MFG equilibrium. Chapters 3 and 4 deal with a MFG model in which producers compete to sell an exhaustible resource such as oil, coal, natural gas, or minerals. In Chapter 3, we propose an alternative approach based on a variational method to formulate the MFG problem, and we explore the deterministic limit (without fluctuations of demand) in a regime where re- sources are renewable or abundant. In Chapter 4 we address the rigorous link between the Cournot MFG model and the N-Player Cournot competition when N is large. In Chapter 5, we introduce a MFG model for the optimal execution of a multi-asset portfolio. We start by formulating the MFG problem, then we compute the optimal execution strategy for a given investor knowing her/his initial inventory and we carry out several simulations. Next, we analyze the influence of the trading activity on the observed intra-day pattern of the covariance matrix of returns and we apply our results in an empirical analysis on a pool of 176 US stocks.