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Abstracts and Talk Materials

Mean field type models describing the limiting behavior of stochastic differential game problems as the number of players tends to infinity, have been recently introduced by J-M. Lasry and P-L. Lions.

They may lead to systems of evolutive partial differential equations coupling a forward Bellman equation and a backward Fokker-Planck equation. The forward-backward structure is an important feature of this system, which makes it necessary to design new strategies for mathematical analysis and numerical approximation.

In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including: existence and uniqueness properties, a priori bounds on the solutions of the discrete schemes, convergence, and algorithms for solving the resulting nonlinear systems of equations. Some numerical experiments are presented.

Collaborators: F. Camilli, I. Capuzzo Dolcetta, V. Perez

They may lead to systems of evolutive partial differential equations coupling a forward Bellman equation and a backward Fokker-Planck equation. The forward-backward structure is an important feature of this system, which makes it necessary to design new strategies for mathematical analysis and numerical approximation.

In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including: existence and uniqueness properties, a priori bounds on the solutions of the discrete schemes, convergence, and algorithms for solving the resulting nonlinear systems of equations. Some numerical experiments are presented.

Collaborators: F. Camilli, I. Capuzzo Dolcetta, V. Perez

Following Lasry and Lions, we present some aspects of the
existence and uniqueness theory for the mean field games systems,
emphasizing the links with optimal control of Hamilton-Jacobi or of
Fokker-Planck equations.

November 12, 2012

We review a series of recent results on Mean Field Games, including existence and the construction of approximate Nash equilibria. We also present the analog of the stochastic maximum principle approach to the optimal control of stochastic dynamics of the McKean-Vlasov type. In both cases existence results are proven by solving forward-backward stochastic differential equations of the McKean-Vlasov type. (joint works with F. Delarue)

In this talk we will discuss a number of techniques to establish existence of smooth solutions of a class of mean-field games which have a variational structure.

We start by showing that a number of mean-field games with local dependence on the player's density can be regarded as Euler-Lagrange equations for certain functionals. These functionals are convex but in a large number of important examples are not coercive. We will then discuss a number of techniques to establish a-priori estimates which when coupled with continuation methods allow to prove the existence of smooth solutions.

We start by showing that a number of mean-field games with local dependence on the player's density can be regarded as Euler-Lagrange equations for certain functionals. These functionals are convex but in a large number of important examples are not coercive. We will then discuss a number of techniques to establish a-priori estimates which when coupled with continuation methods allow to prove the existence of smooth solutions.

Using MFG monotone systems one can push some features (like agents strategies based on recursive anticipations) inside queuing theory. This MFG-queuing methodology will be illustrated through a model of order books dynamics.

Mean field games are everywhere in economics. Why? Because heterogeneity is everywhere. For example, macroeconomists often use heterogeneous agent models to understand the interactions between income and wealth distribution and aggregates like GDP. Classic examples are papers by Aiyagari (1994) and Krusell and Smith (1998). But the mathematical structure of these models is not well understood and numerical solution often resorts to somewhat awkward approximation techniques. These models are really mean field games: Individuals optimize taking as given the evolution of the wealth distribution (HJB equation), and the evolution of the wealth distribution is determined by individual savings behavior (Kolmogorov Forward Equation). I argue that there should be high payoffs from well-trained mathematicians working on issues of numerical solution, existence and uniqueness. I will also present a "Boltzmann mean field game" that came up in some of my own research (Lucas and Moll, 2012).

We discuss the long time average of Mean Field Games systems as the time horizon tends to infinity and the convergence towards a stationary ergodic mean field game, both in case of local and nonlocal coupling in the cost functional.
We also prove that convergence holds at exponential rate, exploiting two completely different approaches; in case of a local, strongly monotone, coupling, we use estimates coming from the Hamiltonian structure of the system, while in case of a nonlocal, regularizing coupling, we use smoothing properties and the exponential decay of the linearized system. Joint works with P. Cardaliaguet, J-M. Lasry and P-L. Lions.