Mean Field Games and Optimal Contracts

金融、经济和工程中的许多问题都涉及到大量代理人之间的竞争或相互作用。传统上，由于系统的庞大规模和交互的博弈性，大人口游戏一直很难分析。平均场博弈理论为近似这些复杂系统提供了一种有用的方法，该理论有助于更好地理解相互作用的影响，以及人口如何对不同的补偿方案或公共政策做出反应。另一方面，2008年的金融危机暴露了金融模式的脆弱性。从那时起，开发考虑模型风险的金融理论的愿望大大增强。这项研究项目探索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玩家的动态比赛，其中玩家之间的互动是通过各自项目完成时间的排名来进行的。该模型适用于许多公司或个人争先恐后地实现目标的情况。目标是了解均衡，并设计一种奖励方案，在组织者预算有限的情况下鼓励项目早日完成，或者在期望的完成率下将预算降至最低。该项目的第二部分涉及平均场游戏近似的精度，当在许多应用中，典型的竞争规模只是适度大的。目的是研究平均场博弈近似在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当参与者之间的互动既不是通过状态过程也不是通过成本结构，至少不是以直接的方式，而是通过停止行为所揭示的信念或信息时，最优停止的平均场博弈。该项目的最后部分考虑了在交易成本和模型不确定性下的金融市场中未定权益的定价和对冲，其中模型的不确定性用一组概率度量来描述。一般来说，集合不需要具有支配所有其他度量的参考度量，因此，不能应用来自函数分析的标准工具，并且需要新的技术。

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.