BERTUCCI Charles

We present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under standard assumptions. This notion allows us to work with solutions which are merely continuous in the measure argument, in the case of first order master equations. We study several structures of common noises, in particular ones in which common jumps (or aggregate shocks) can happen randomly, and ones in which the correlation of randomness is carried by an additional parameter.

This paper presents a novel approach to characterize the dynamics of the limit spectrum of large random matrices. This approach is based upon the notion we call "spectral dominance". In particular, we show that the limit spectral measure can be determined as the derivative of the unique viscosity solution of a partial integro-differential equation. This also allows to make general and "short" proofs for the convergence problem. We treat the cases of Dyson Brownian motions, Wishart processes and present a general class of models for which this characterization holds.

In this paper, we present an extension of Uzawa's algorithm and apply it to build approximating sequences of mean field games systems. We prove that Uzawa's iterations can be used in a more general situation than the one in it is usually used. We then present some numerical results of those iterations on discrete mean field games systems of optimal stopping, impulse control and continuous control.

This paper is interested in the description of the density of particles evolving according to some optimal policy of an impulse control problem. We first fix sets on which the particles jump and explain how we can characterize such a density. We then investigate the coupled case in which the underlying impulse control problem depends on the density we are looking for : the mean field games of impulse control. In both cases, we give a variational characterization of the densities of jumping particles.

We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We prove first results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. We focus in this paper in the finite state space case.

This is a work in progress. The aim is to propose a plausible mechanism for the short term dynamics of the oil market based on the interaction of economic agents. This is a theoretical research which by no means aim at describing all the aspects of the oil market. In particular, we use the tools and terminology of game theory, but we do not claim that this game actually exists in the real world. In parallel, we are currently studying and calibrating a long term model for the oil industry, which addresses the interactions of a monopolists with a competitive fringe of small producers. It is the object of another paper that will be available soon. The present premiminary version does not contain all the economic arguments and all the connections with our long term model. It mostly addresses the description of the model, the equations and numerical simulations focused on the oil industry short term dynamics. A more complete version will be available soon.

We present results of existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions.

This note is concerned with a modeling question arising from the mean field games theory. We show how to model mean field games involving a major player which has a strategic advantage, while only allowing closed loop markovian strategies for all the players. We illustrate this property through three examples.

We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of " noise " in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.

This paper is interested in the description of the density of particles evolving according to some optimal policy of an impulse control problem. We first fix sets on which the particles jump and explain how we can characterize such a density. We then investigate the coupled case in which the underlying impulse control problem depends on the density we are looking for : the mean field games of impulse control. In both cases, we give a variational characterization of the densities of jumping particles.

In this paper, we present an extension of Uzawa's algorithm and apply it to build approximating sequences of mean field games systems. We prove that Uzawa's iterations can be used in a more general situation than the one in it is usually used. We then present some numerical results of those iterations on discrete mean field games systems of optimal stopping, impulse control and continuous control.

This thesis is concerned with new models of mean field games. First, we study models of optimal stopping and impulse control in the case when there is no common noise. We build an appropriate notion of solutions for those models. We prove the existence and the uniqueness of such solutions under natural assumptions. Then, we are interested with several properties of mean field games. We study the limit of such models when the anticipation of the players vanishes. We show that uniqueness holds for strongly coupled mean field games (coupled via strategies) under certain assumptions. We then prove some regularity results for the master equation in a discrete state space case with common noise. We continue by giving a generalization of Uzawa’s algorithm and we apply it to solve numerically some mean field games, especially optimal stopping and impulse control problems. The last chapter presents an application of mean field games. This application originates from problems in telecommunications which involve a huge number of connected devices.

This thesis deals with the study of new medium field game models. We first study optimal stopping and impulse control models in the absence of common noise. We construct for these models a notion of adapted solution for which we prove existence and uniqueness results under natural assumptions. Then, we focus on several properties of mean-field games. We study the limit of these models to pure evolution models when the players' anticipation tends to 0. We show the uniqueness of equilibria for strongly coupled systems (coupled by strategies) under certain assumptions. We then prove some regularity results on a master equation that models a mean field game with common noise in a discrete state space. We then present a generalization of the standard Uzawa algorithm and apply it to the numerical solution of some mean-field game models, in particular optimal stopping or impulse control. Finally, we present a concrete case of mean-field game that comes from problems involving a large number of connected devices in telecommunications.

No summary available.

We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of " noise " in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.