Mean-Field games with absorption and singular controls

The first part of the work is devoted to mean-field games with absorption, a class of games that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit a given boundary. In most of the literature on mean-field games, all players stay in the game until the end of the period, while in many applications, especially in economics and finance, it is natural to have a mechanism deciding when a player has to leave. Such a mechanism can be modelled by introducing an absorbing boundary for the state space. The second part of the thesis, deals with mean-field games of finite-fuel capacity expansion with singular controls. While singular control problems with finite (and infinite) fuel find numerous applications in the economic literature and originated from the engineering literature in the late 60’s, many-player game versions of these problems have only very recently been introduced. They are a natural extension of the single agent set-up and allow to model numerous applied situations. In our work in particular, we make assumptions on the structure of the interaction across players that are suitable to model the so-called goodwill problem. Altogether, the original contribution to the mean-field games literature of the present work is threefold. First, it contributes to the development of mean-field games with absorption, continuing the work of Campi and Fischer (2018) and considerably generalizing the original model by relaxing the assumptions and setting it into a more abstract, infinite-dimensional, framework. Second, it introduces a new set of tools to deal with mean-field games with singular controls, extending the well-known connection between singular stochastic control and optimal stopping to mean-field games. Finally, it also contributes to the numerical literature on mean-field games, by proposing a numerical scheme to approximate the solutions of mean-field games with singular controls with a constructive approach. Overall, this thesis focuses on newly introduced branches of the theory of meanfield games that display a high potential for economic and financial applications, contributing to the literature not only by further developing the existing theory but also by working in directions that make the these models more suitable to applications