CAREER: Mean Field Games with Economics Applications: New Techniques in Partial Differential Equations

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Mean field games are mathematical models of large-scale interactions between rational agents. They can be used to explain complex phenomena in the economy, such as business cycles and inequality, but we do not fully understand the mathematical theory behind them. The project will develop new mathematical techniques in the field of differential equations to determine the theoretical properties of mean field games that arise in economics. This research will contribute to a rational discussion of economics, and by extension public policy, by determining which mathematical models are viable and how their solutions behave. Moreover, the research is integrated with an interdisciplinary education and outreach plan, engaging the public by showing the connection between mathematics and social dynamics. The project will also develop courses and seminars that use tested practices to integrate mathematics instruction with the social sciences, with the cooperation of instructors in economics and political science. Both research and education aspects of the project will provide training to students at graduate and undergraduate levels, thus contributing to the development of a diverse, globally competitive STEM workforce. Mean field game theory provides useful mathematical models in economics using coupled systems of partial differential equations, but their analysis is rendered difficult by the presence of nonlocal and stochastic interactions. This project will develop new techniques for partial differential equations to determine whether these models are well-posed. The first objective is to prove new results for forward-backward coupled systems of nonlinear, integro-differential equations that describe macroeconomic phenomena. The second objective is to prove new results for infinite dimensional equations that model economic shocks. To achieve these objectives, the principal investigator will develop new analytical techniques based on recent advances in nonlocal equations, forward-backward systems, and the Master Equation for mean field games.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.