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

Summary 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.