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

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。随机微分对策是指在随机、不确定的环境中，多个参与者相互作用的最优决策问题。在这样的游戏中，参与者试图单独或集体地优化给定的目标，而他们的决定会影响他们的同伴。这种游戏在我们周围很普遍。例如，在金融经济学中，当考虑大量银行从事银行间借贷的系统性违约风险时，在城市规划中，当模拟通勤者试图找到最短路径同时避免拥挤时，或者在流行病学中，当所有社会成员聚集在一起以减少病毒的传播时。在生物学、经济学和工程学中也可以找到几个例子。当人口规模变得很大时，（随机）微分博弈变得非常棘手，并导致严重的分析和计算挑战。一个基本的数学启发式表明，当人口规模足够大时，它足以分析代表整个人口的“典型”或平均玩家的行为。该项目的目标是开发数学技术，使人们能够使这种启发式的严格，并了解其范围和后果，因为它们涉及到计算问题和应用程序中的金融建模的泡沫形成和投资之间的竞争代理。研究生和本科生都将参与这项工作。本研究计划将建立一个严格而系统的框架，通过纯粹的概率论证来理解随机微分博弈中的平均场博弈极限，同时解释理论根源的物理（热力学）直觉。作为实现其目标的主要工具，该项目将介绍和分析一种新的形式的混沌传播的相互作用粒子系统的时间发展，并在此设置的功能不等式。这种新的方法将有有趣的后果，因为它将允许提供大偏差原则和非渐近收敛速度的平均场限制的竞争以及合作游戏。我们还将考虑这些结果的后果，因为它们与数值模拟和定量金融建模中的应用有关。拟议的研究奋进不仅将推进目前活跃的平均场游戏领域，但在一般的相互作用的粒子系统的研究将是极大的兴趣。我们可以预见，混沌的反向传播和用于分析经验过程收敛性的技术将得到广泛的应用。例如，我们将采用这种技术来分析（大规模）最优运输问题，并估计金融风险措施。事实上，我们将开发无模型、完全数据驱动的方法，用于基于经验过程理论和混沌传播的一般凸风险度量的估计。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.