Courses and Program

Courses description

Giorgio Ferrari

Mean Field Games: From Foundations to Singular Controls and Optimal Stopping

This short course provides an introduction to mean field games, with particular emphasis on their probabilistic formulation and on models involving singular control and optimal stopping. We will start from the basic idea of mean field interaction: a representative agent optimizes against the distribution of a large population, while equilibrium requires consistency between the agent’s optimal behavior and the induced population flow. After discussing this fixed point structure and some analytical tools, we will focus on classes of models motivated by economics and finance.

Special attention will be devoted to mean field games in which agents make irreversible, monotone, or stopping decisions, both in stationary and non-stationary settings. In these models, a central theme is the interplay between free-boundary methods, reflected or absorbed diffusions, and fixed point arguments in the characterization of equilibria. We will also discuss how structural properties of the interaction, such as monotonicity or submodularity, can be exploited in suitable models to obtain comparison results and to gain insights into uniqueness or multiplicity of equilibria. The course will conclude by highlighting connections between mean field games and mean field control problems, as well as recent perspectives related to learning-based approaches.

Huyen Pham

Mean-Field Control: Theory, Methods, from Foundations to Learning

These lectures cover the optimal control of McKean-Vlasov equations, a topic that has attracted growing interest in connection with mean-field game theory, large population stochastic control, and more recently machine learning.

We start with mean-field Markov decision processes in discrete time, where the key ideas — propagation of chaos, lifting to the Wasserstein space, dynamic programming — can be introduced transparently. We then develop the continuous-time theory of controlled McKean-Vlasov SDEs, building the necessary tools: differentiation with respect to probability measures, Itô's formula along flows of marginal laws, the Master Bellman equation, and the stochastic maximum principle with its associated FBSDE system. Linear-quadratic problems serve as a running illustration throughout.

The last part of the course is devoted to numerical methods and applications. We present neural network-based algorithms and reinforcement learning approaches for solving mean-field control problems on the Wasserstein space, and discuss recent extensions to  non-exchangeable mean-field control for heterogeneous interacting systems. We conclude with an application to generative modeling, where learning  stochastic dynamics from distributional observations is formulated as a McKean-Vlasov control problem, with optimality conditions given by a tractable FBSDE system.