Mean Field Games and Optimal Contracts

金融、经济和工程中的许多问题都涉及大量主体之间的竞争或相互作用。由于庞大的系统规模和互动的游戏性质，大型人口游戏通常很难分析。平均场博弈理论提供了一种有用的方法来近似这些复杂的系统，该理论有助于提高对相互作用的影响以及种群如何对不同的补偿方案或公共政策作出反应的理解。另一方面，2008年金融危机暴露了金融模式的脆弱性。从那时起，发展考虑到模型风险的金融理论的需求急剧增长。本研究项目探讨平均场博弈论和稳健金融中的数学问题。项目的第一部分分析了一个涉及大量参与者的动态竞争，其中参与者之间的互动是通过对各自项目的完成时间进行排名。该模型适用于许多公司或个人为了第一个实现目标而竞争的情况。目标是理解平衡并设计奖励方案，鼓励组织者在有限的预算下尽早完成项目，或者在期望的完成率下最小化预算。该项目的第二部分关注的是平均场博弈近似的准确性，而在许多应用中，典型的竞争规模只有适度大。目的是研究平均场博弈近似的水动力极限附近的波动，以提高精度。项目的第三部分研究了参与者之间的交互既不是通过状态过程，也不是通过成本结构，至少不是通过直接的方式，而是通过停止行动所揭示的信念或信息进行交互的最优停止的平均场博弈。项目的最后一部分考虑了交易成本和模型不确定性下金融市场或有债权的定价和对冲，其中模型不确定性由一系列概率度量来描述。一般来说，这个集合不需要有一个支配所有其他度量的参考度量，因此，不能应用功能分析的标准工具，而需要新的技术。

Many questions in finance, economics, and engineering involve competition or interaction among a large number of agents. Large-population games have traditionally been difficult to analyze due to the large size of the system and the game nature of the interaction. The theory of mean field games provides a useful way to approximate these complex systems, and the theory helps to improve understanding of the effects of interactions and how populations react to different compensation schemes or public policies. On the other hand, the 2008 financial crisis revealed the fragility of financial models. Since then, the urge to develop financial theories that take into account model risk has grown tremendously. This research project explores mathematical questions in mean field game theory and robust finance. The first part of the project analyzes a dynamic competition involving a large number of players, where the interaction among players is through the ranking of the completion time of their respective projects. The model applies to situations where many firms or individuals compete to be the first to achieve a goal. The objective is to understand the equilibrium and design a reward scheme that encourages early project completion given that the organizer has a limited budget, or to minimize the budget given a desired rate of completion. The second part of the project is concerned with the accuracy of the mean field game approximation, when in many applications, the typical size of competition is only modestly large. The objective is to study the fluctuation around the hydrodynamic limit of the mean field game approximation so as to improve accuracy. The third part of the project studies a mean field game of optimal stopping when the interaction among players is neither through the state process nor the cost structure, at least not in a direct way, but through the belief or information revealed from the action of stopping. The last part of the project considers pricing and hedging of contingent claims in a financial market under both transaction costs and model uncertainty, where model uncertainty is described by a collection of probability measures. In general the collection need not have a reference measure that dominates every other measure, and therefore, standard tools from functional analysis cannot be applied, and new techniques are called for.