Economic Models in MFG and Related Topics - Part 2

Day one: Wednesday 19 March 2025

Please note the morning session will take place at the Institute of Advanced Study (IAS), Cosin's Hall, Palace Green. The afternoon session will take place in the department of Mathematical Sciences

Day 2: Thursday, 20 March 2025 (IAS Seminar Room)

Contact Dr Alpár Mészáros, Mathematical Sciences (alpar.r.meszaros@durham.ac.uk) or Dr Mauro Bambi, Economics (mauro.bambi@durham.ac.uk) to express interest in attending.  

_____________

Abstracts and details are noted below.

Dr Marta Leocata (LUISS University, Rome) 
Title: A Mean Field Game approach for pollution regulation of competitive firms 
Abstract:  We develop a model based on mean-field games of competitive firms producing similar goods according to a standard AK model with a depreciation rate of capital generating pollution as a byproduct. Our analysis focuses on the widely-used cap-and-trade pollution regulation. Under this regulation, firms have the flexibility to respond by implementing pollution abatement, reducing output, and participating in emission trading, while a regulator dynamically allocates emission allowances to each firm. The resulting mean-field game is of linear quadratic type and equivalent to a mean-field type control problem. We find explicit solutions to this problem through the solutions to differential equations of Riccati type. Further, we investigate the carbon emission equilibrium price that satisfies the market clearing condition and find a specific form of FBSDE of McKean-Vlasov type with common noise. The solution to this equation provides an approximate equilibrium price.

Gabriele Bolli (Sapienza University, Rome) 
Title: Regularizing effects of Ornstein-Uhlenbeck semigroups in infinite dimension: application to optimal control of stochastic PDEs
Abstract: We study a class of stochastic optimal control problems with infinite horizon and unbounded control operator using the dynamic programming approach. Our analysis is related to the regularizing properties of the Ornstein-Uhlenbeck semigroup in separable Hilbert spaces. By applying a partial smoothing result, we prove the existence and uniqueness of solutions to the associated Hamilton-Jacobi-Bellman equations. These results can be applied to the analysis of boundary control problems and stochastic delay equations. Such problems are relevant in economics, for instance, in studying spatial growth dynamics, optimal advertising and optimal growth models with vintage capital.

Dr Quentin Petit (EDF, France) 
Title: Growth Model with Externalities for the Energy Transition via MFG with a Common External Variable 
Abstract: TBA

Professor Nizar Touzi (New York University) 
Title: Optimal transport methods in risk management  
Abstract: We review several optimal transport problems motivated by risk management in financial engineering and optimal incentive theory in economics, with interesting connections to the Skorohod Embedding Problem in probability, mean field games in optimal control theory and generative methods in artificial intelligence. We finally discuss important applications to model risk measurement and management.

Day 2

Dr Daria Ghilli (University of Pavia) 
Title: Mean Field Games in Hilbert spaces and Applications to Economics
Abstract:  We study a class of linear quadratic Mean Field Games (MFG) in infinite dimension, where the state variable lives in a Hilbert space. In the class of problems we study the state equation is a delay equation or a PDE, which can be written as an ODE in a suitable Hilbert space. We study the case, considered in most finite dimensional contributions on the topic, where the dependence on the distribution enters just in the objective functional through the mean. This feature allows to reduce the usual mean field game system to a Riccati equation and a forward-backward coupled system of abstract evolution equations. Such system is completely new in infinite dimension and no results have been proved on it so far. We show existence and uniqueness of solutions for such system, applying a delicate approximation procedure. Moreover we study the Nash system and the Master equation associated, and prove the convergence of the solution of the Nash system to the solution of the Master equation. Finally, we apply these results to a vintage capital model where the state equation for the capital is a first order PDE and to a production output planning problem with delay in the control variable.

Dr Giovanni Zanco (University of Siena) 

Title: A mean-field path-dependent model for life-cycle optimal investment

Abstract: I will introduce a simple model of optimal portfolio choice for wealth allocation in the presence of labor income dynamics that are both path-dependent and mean-field. Reformulating the optimization problem in terms of control of a McKean-Vlasov equation, I will provide the explicit solution in a simple case through the analysis of an infinite-dimensional Hamilton-Jacobi-Bellman equation. Such equation arises in two equivalent ways, one obtained by considering a suitable additional variable for the problem and the other resulting instead from the direct application of optimization methods in Wasserstein spaces. I will then discuss possible extensions of the model together with some open problems.

Dr Mohamed Bahlali (Aix-Marseille University)  
Title: A Mean-Field Game Model of Trade and Migration  
Abstract: TBA

Professor Fausto Gozzi (LUISS University, Rome) 
Title: On some Mathematical Models for Biodiversity and Agroecology
Abstract: In this talk we present some ideas on how to model the evolution of biodiversity in the Antropocene, (the era where the presence of homo sapiens is modifying deeply the biosphere) and its relations with the economic variables.  In particular we focus on two aspects:

- on one side the relations of biodiversity loss, deforestation and agriculture in an optimal control model;

- on the other side the space-time evolution of biodiversity in a spatial mean field game.

Papers in progress with E. Augeraud, R. Boucekkine, A. Calvia, D. Ghilli, M. Leocata, F. Masiero