Approximation of deterministic mean field games under polynomial growth conditions on the data

We consider a deterministic mean field games problem in which a typical agent solves an optimal control problem where the dynamics is affine with respect to the control and the cost functional has a growth which is polynomial with respect to the state variable. In this framework, we construct a mean field game problem in discrete time and finite state space that approximates equilibria of the original game. Two numerical examples, solved with the fictitious play method, are presented.Comment: 14 pages, 3 figure

A semi-discrete approximation for a first order mean field game problem

In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [18]. Its solution (v;m) can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [18]). We propose a semi-discrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to (v;m) when the discretization parameter tends to zero. © American Institute of Mathematical Sciences

Nonlinear PDEs in ergodic control, Mean Field Games and prescribed curvature problems

This thesis is concerned with nonlinear elliptic PDEs and system of PDEs arising in various problems of stochastic control, differential games, specifically Mean Field Games, and differential geometry. It is divided in three parts. The first part is focused on stochastic ergodic control problems where both the state and the control space is R^d. The interest is in giving conditions on the fixed drift, the cost function and the Lagrangian function that are sufficient for synthesizing an optimal control of feedback type. In order to obtain such conditions, an approach that combines the Lyapunov method and the approximation of the problem on bounded sets with reflection of the diffusions at the boundary is proposed. A general framework is developed first, and then particular cases are considered, which show how Lyapunov functions can be constructed from the solutions of the approximating problems. The second part is devoted to the study of Mean Field Games, a recent theory which aims at modeling and analyzing complex decision processes involving a very large number of indistinguishable rational agents. The attention is given to existence results for the multi- population MFG system of PDEs with homogeneous Neumann boundary conditions, that are obtained combining elliptic a-priori estimates and fixed point arguments. A model of segregation between human populations, inspired by ideas of T. Schelling is then proposed. The model, that fits into the theoretical framework developed in the thesis, is analyzed from the qualitative point of view using numerical finite-difference techniques. The phenomenon of segregation between the population densities arises in the numerical experiments on the particular mean field game model, assuming mild ethnocentric attitude of people as in the original model of Schelling. In the last part of the thesis some results on existence and uniqueness of solutions for the prescribed k-th principal curvature equation are presented. The Dirichlet problem for such a family of degenerate elliptic fully nonlinear partial differential equations is solved using the theory of Viscosity solutions, by implementing a version of the Perron method which involves semiconvex subsolutions; the restriction to this class of functions is sufficient for proving a Lipschitz estimate on the elliptic operator with respect to the gradient entry which is also required for obtaining the comparison principle. Existence and uniqueness are stated under general assumptions, and examples of data which satisfy the general hypotheses are provided