CAREER: Quantitative Approach to Large-population Stochastic Dynamic Games

Nadtochiy1651294存在着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(代理人)相互作用的产物，他们达到的平衡反映了他们的目标，以优化他们的个人目标。如果智能体在不确定的环境中动态地做出决策，那么这种相互作用的结果可以方便地用一个连续体-博弈者随机动态博弈来描述。研究人员追求几个研究方向，统一于建立博弈论模型的想法，这些模型一方面以足够的精度捕捉真实世界的现象，另一方面允许对均衡的易于处理的表示。这样的模型可以用来建立量化结果，即对系统未来的演变做出预测，或优化系统中的相互作用规则。对这类游戏的研究受到各种应用的推动。作为特例，研究者考虑了市场微观结构和系统性风险产生的模型。该项目的结果可用于预测金融市场的潜在不稳定性，以及为金融交易所和银行系统制定有效的监管政策。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，从而产生了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，更加强调数学在金融和经济问题中的新应用。研究生被包括在该项目的工作中。虽然在动态随机博弈中有许多构建均衡的抽象数学结果，但这些结果依赖于往往与现实模型不相容的假设。调查者考虑了几类自然无法满足标准假设的大人口博弈，要么是由于代理人策略中存在停止时间，要么是因为代理人之间的单一类型的相互作用。在这两种情况下，由于缺乏所需的连续性或单调性，标准不动点结果不能直接应用于构造平衡点。研究人员研究了几种克服这些困难的方法，这些方法导致了数学问题本身就很有趣。本研究对连续体博弈(包括但不限于平均场博弈)、最优随机控制和混沌传播理论的研究具有重要意义。具体地说，这个项目解决了具有斜反射的倒向随机微分方程理论中的具体问题，以及通过命中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。研究生被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.