On the system of partial differential equations arising in mean field type control

We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations arising from the finite horizon control of McKean-Vlasov dynamics. We give examples of existence and uniqueness results. Finally, we propose some simple models for the motion of pedestrians and report about numerical simulations in which we compare mean filed games and mean field type control

Mean field type control with congestion

We analyze some systems of partial differential equations arising in the theory of mean field type control with congestion effects. We look for weak solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as the optima of two optimal control problems in duality

Multi-population Mean Field Games with Multiple Major Players: Application to Carbon Emission Regulations

In this paper, we propose and study a mean field game model with multiple populations of minor players and multiple major players, motivated by applications to the regulation of carbon emissions. Each population of minor players represent a large group of electricity producers and each major player represents a regulator. We first characterize the minor players equilibrium controls using forward-backward differential equations, and show existence and uniqueness of the minor players equilibrium. We then express the major players' equilibrium controls through analytical formulas given the other players' controls. Finally, we then provide a method to solve the Nash equilibrium between all the players, and we illustrate numerically the sensitivity of the model to its parameters

Deep Learning for Population-Dependent Controls in Mean Field Control Problems

In this paper, we propose several approaches to learn optimal population-dependent controls, in order to solve mean field control problems (MFC). Such policies enable us to solve MFC problems with generic common noise. We analyze the convergence of the proposed approximation algorithms, particularly the N-particle approximation. The effectiveness of our algorithms is supported by three different experiments, including systemic risk, price impact and crowd motion. We first show that our algorithms converge to the correct solution in an explicitly solvable MFC problem. Then, we conclude by showing that population-dependent controls outperform state-dependent controls. Along the way, we show that specific neural network architectures can improve the learning further.Comment: 21 pages, 6 figure