CAREER: A new form of propagation of chaos and its applications to large population games and risk management

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Stochastic differential games are optimal decision problems involving several players in interaction, acting in a random, uncertain environment. In such games players try either individually or collectively to optimize a given objective, while their decisions influence that of their peers. This type of game is widespread around us. For instance, in financial economics when considering the systemic risk of default by a large number of banks engaged in inter-bank borrowing and lending, in urban planning when modeling commuters trying to find the shortest path while avoiding congestions, or in epidemiology when all members of a society come together to reduce the spread of a virus. Several examples can also be found in biology, economics, and engineering. When the size of the population becomes large, (stochastic) differential games become notoriously intractable, and cause serious analytical and computational challenges. A basic mathematical heuristic suggests that when the size of the population is sufficiently large, it suffices to analyze the behavior of a "typical" or average player that represents the entire population. The goal of the proposed project is to develop mathematical techniques allowing one to make this heuristic rigorous and to understand its scope and consequences as they relate to computational issues and applications in the financial modeling of bubble formation and investment among competitive agents. Both graduate and undergraduate students will be involved in this work. An extensive outreach program helping to increase the participation of minorities in engineering graduate school will be established.The present research project will lay down a rigorous and systematic framework for understanding the mean field game limit in stochastic differential games by purely probabilistic arguments, while explaining the physical (thermodynamic) intuition at the root of the theory. As the main tool to achieve its objectives, this project will introduce and analyze a new form of propagation of chaos for interacting particle systems evolving backward in time, and functional inequalities in this setting. This novel approach will have interesting consequences as it will allow to provide both large deviation principles and non-asymptotic convergence rates to the mean field limit for competitive as well as cooperative games. We will also consider the consequences of these results as they relate to numerical simulations and applications in quantitative financial modeling. The proposed research endeavor will not only advance the currently active area of mean field games but will be of great interest in the study of interacting particle systems in general. We foresee that backward propagation of chaos and the techniques used to analyze convergence of empirical processes will find numerous applications. For instance, we will employ such techniques to analyze (large scale) optimal transportation problems, and to the estimation of financial risk measures. In fact, we will develop model–free, fully data-driven approaches for the estimation of general convex risk measures based on empirical process theory and propagation of chaos.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.