## A mean-field game model of economic growth : an essay in regularity theory

##### Abstract

In this thesis, we present a priori estimates for solutions of a mean-field game (MFG) defined
over a bounded domain Ω ⊂ ℝd. We propose an application of these results to a model of capital
and wealth accumulation.
In Chapter 1, an introduction to mean-field games is presented. We also put forward some of
the motivation from Economics and discuss previous developments in the theory of differential
games. These comments aim at indicating the connection between mean-field games theory, its
applications and the realm of Mathematical Analysis.
In Chapter 2, we present an optimal control problem. Here, the agents are supposed to be
undistinguishable, rational and intelligent. Undistinguishable means that every agent is governed
by the same stochastic differential equation. Rational means that all efforts of the agent is to
maximize a payoff functional. Intelligent means that they are able to solve an optimal control
problem. Once we describe this (stochastic) optimal control problem, we produce a heuristic
derivation of the mean-field games system, which is summarized in a Verification Theorem; this
gives rise to the Hamilton-Jacobi equation (HJ). After that, we obtain the Fokker-Plank equation
(FP). Finally, we present a representation formula for the solutions to the (HJ) equation, together
with some regularity results.
In Chapter 3, a specific optimal control problem is described and the associated MFG is
presented. This MFG is prescribed in a bounded domain
Ω ⊂ ℝd, which introduces substantialadditional challenges from the mathematical view point. This is due to estimates for the solutionsat the boundary in Lp. The rest of the chapter puts forward two well known tips of estimates: theso-called Hopf-Lax formula and the First Order Estimate.
In Chapter 4, the wealth and capital accumulation mean-field game model is presented. The
relevance of studying MFG in a bounded domain then becomes clear. In light of the results obtained
in Chapter 3, we close Chapter 4 with the Hopf-Lax formula, and the First Order estimates.
Three appendices close this thesis. They gather elementary material on Stochastic Calculus
and Functional Analysis.