CAREER: Quantitative Approach to Large-population Stochastic Dynamic Games

存在各种各样的自然和社会现象，在这些现象中，可观察到的结果是大量参与者（主体）之间相互作用的产物，这些参与者（主体）达到了一种反映他们优化个人目标的平衡。如果智能体在不确定的环境中动态地做出决策，则这种交互的结果可以方便地用连续参与人随机动态博弈来描述。研究者追求几个研究方向，统一了建立博弈论模型的想法，一方面以足够的精度捕获现实世界的现象，另一方面允许对平衡的易于处理的表示。这样的模型可以用来建立定量的结果，即对系统的未来演变做出预测，或者优化系统中相互作用的规则。研究这类游戏的动机是各种各样的应用。作为特殊案例，研究者考虑了市场微观结构和系统风险产生的模型。该项目的结果可用于预测金融市场的潜在不稳定性，并为金融交易所和银行系统设计有效的监管政策。此外，本项目中分析的系统类别包括经济学、神经科学和社会学中的其他相关模型，从而产生大量可能造福社会的潜在应用。该项目的教育部分包括为本科生和研究生设计教材，更强调数学在金融和经济问题中的新应用。研究生也包括在该项目的工作中。虽然有许多抽象的数学结果用于构建动态随机博弈的均衡，但这些结果依赖于通常与现实模型不相容的假设。研究者考虑了几类自然不能满足标准假设的大群体博弈，要么是因为代理策略中存在停止时间，要么是因为代理之间的交互类型单一。在这两种情况下，由于缺乏必要的连续性或单调性，标准不动点结果不能直接用于构造平衡。研究者研究了几种方法来克服这些困难，这些困难本身就会导致有趣的数学问题。本研究对连续人博弈（包括但不限于平均场博弈）、最优随机控制和混沌传播理论做出了贡献。特别地，本项目解决了斜反射倒向随机微分方程理论中的具体问题，以及通过碰撞时间具有奇异相互作用的粒子系统的极限行为问题。这一分析结果为基于均衡的建模提供了新的工具。研究生也包括在该项目的工作中。

Nadtochiy1651294 There exists a wide variety of natural and social phenomena in which the observable outcome is a product of interactions among a large number of participants (agents), who reach an equilibrium that reflects their aims to optimize their individual objectives. If the agents make their decisions dynamically, in an uncertain environment, the outcome of such interactions can be conveniently described by a continuum-player stochastic dynamic game. The investigator pursues several research directions, unified by the idea of establishing the game-theoretic models, which on the one hand capture the real-world phenomena with sufficient precision, and on the other hand allow for tractable representations of the equilibria. Such models can be used to establish quantitative results, i.e., to make predictions about the future evolution of a system, or to optimize the rules of interaction in the system. The study of such classes of games is motivated by various applications. As particular cases, the investigator considers models arising in market microstructure and systemic risk. The results of the project can be used for predicting the potential instabilities in financial markets and for designing effective regulatory policies for financial exchanges and banking systems. In addition, the class of systems analyzed in this project includes other relevant models in economics, neuroscience, and sociology, resulting in a large array of potential applications that may benefit society. The educational component of the project includes designing teaching materials, for both undergraduate and graduate students, with stronger emphasis on the novel applications of mathematics to problems of finance and economics. Graduate students are included in the work of the project. While there are many abstract mathematical results for constructing equilibria in dynamic stochastic games, these results rely on assumptions that often are incompatible with realistic models. The investigator considers several classes of large-population games that naturally fail to satisfy the standard assumptions, either due to the presence of stopping times in the agents' strategies, or because of the singular type of interactions between the agents. In both cases, the standard fixed-point results cannot be applied directly to construct an equilibrium, due to the lack of required continuity or monotonicity properties. The investigator studies several methods to overcome these difficulties, which lead to mathematical problems interesting in their own right. This study contributes to the theory of continuum-player games (including, but not limited to, mean field games), optimal stochastic control, and the propagation of chaos. In particular, this project addresses specific problems in the theory of backward stochastic differential equations with oblique reflection, as well as the questions of limiting behavior of particle systems with singular interaction through hitting times. The results of this analysis provide new tools for equilibrium-based modeling. Graduate students are included in the work of the project.