MOUZOUNI Charafeddine

Mean-field game systems (MFG) describe equilibrium configurations in differential games with an infinite number of infinitesimal agents. This thesis is structured around three different contributions to the theory of mean-field games. The main goal is to explore applications and extensions of this theory, and to propose new approaches and ideas to deal with the underlying mathematical issues. The first chapter first introduces the key concepts and ideas that we use throughout the thesis. We introduce the MFG problem and briefly explain the asymptotic connection with N-player differential games when N → ∞. We then present our main results and contributions. Chapter 2 explores an MFG model with a non-anticipatory interaction mode (myopic players). Unlike classical MFG models, we consider less rational agents who 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 coupled PDE system resulting from this model, and establish the rigorous link with the corresponding N-Players game. We show that the population of agents can self-organize through a self-correcting process and converge exponentially fast to a well-known MFG equilibrium configuration. Chapters 3 and 4 concern the application of the MFG theory to the modeling of production and marketing processes of products with exhaustible resources (e.g. fossil fuels). In Chapter 3, we propose a variational approach for the study of the corresponding MFG system and analyze the deterministic limit (without demand fluctuations) in a regime where resources are renewable or abundant. In Chapter 4 we treat the MFG approximation by analyzing the asymptotic link between the N-player Cournot model and the MFG Cournot model when N is large. Finally, Chapter 5 considers an MFG model for the optimal execution of a portfolio of assets in a financial market. We explain our MFG model and analyze the resulting PDE system, then we propose a numerical method to compute the optimal execution strategy for an agent given its initial inventory, and present several simulations. Furthermore, we analyze the influence of trading activity on the intraday variation of the covariance matrix of asset returns. Next, we verify our findings and calibrate our model using historical trading data for a pool of 176 US stocks.

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.