Mean Field Games, Information Design and Evolutionary Finance

平均场博弈(MFGs)为大群体随机博弈提供了一种有用的近似，其中玩家通过经验分布耦合在一起。许多曾经由于高维度而难以处理的金融和经济模型现在可以使用最惠国集团的方法进行分析。研究建议的一部分研究了数学、金融和经济学中出现的两个MFG问题。第一个是首席研究员先前关于排名游戏和最优奖励设计工作的继续，我们建议研究一个多阶段平均场比赛，奖励流取决于每个阶段完成时间的排名或进展过程的运行排名。该模型适用于许多公司或个人争先恐后地实现具有多个里程碑的目标的情况。第二种是最优停止的MFG，其中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，至少不是以直接的方式，而是通过停止动作所揭示的信念或信息。所提出的最优停止问题包括对金融事件发生的最快检测和对未知市场变量的值的序贯检验。通过指定每个代理人如何处理公共信息，人们可能能够分析顺应民意对人们决策的影响，以及它是否有利于个人和集体。 该提案还包含两个新的研究方向。一类是随机控制和博弈，其中控制变量是由sigma-代数建模的信息，性能准则涉及给定该信息的随机变量的条件期望。这样的问题被称为贝叶斯说服或信息设计，它在金融、政治科学、执法、医学测试等领域有着广泛的应用。我们的目标是发展静态和动态信息设计问题的数学理论和数值算法，可能有信息约束，以及静态信息博弈。另一项研究将基于选择和突变原则的进化和学习的生物学模型纳入金融市场。这两个主题都将经济学和生物学等其他领域的有趣问题和想法带到了随机控制和数学金融界，这将为新的理论和应用打开大门。

Mean field games (MFGs) provide a useful approximation for large population stochastic games in which the players are coupled through their empirical distribution. Many financial and economic models that were once intractable due to high dimensionality can now be analyzed using the MFG approach. Part of the research proposal studies two MFG problems arisen in mathematical finance and economics. The first one is a continuation of the principal investigator's previous work on rank-based games and optimal reward design, where we propose to look at a multi-stage mean field competition with a reward stream that depends on the ranking of the completion time of each stage or the running rank of the progress process. The model applies to situations where many firms or individuals compete to be the first to achieve a goal with multiple milestones. The second one is a MFG of optimal stopping where 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 optimal stopping problems proposed include the quickest detection of the occurrence of a financial event and the sequential testing of the value of an unknown market variable. By specifying how each agent process the public information, one may be able to analyze the effect of conformity to the public opinion on people's decision making, and whether it benefits people individually as well as collectively. The proposal also contains two new lines of research. One is concerned with stochastic control and games where the control variable is the information, modelled by sigma-algebras, and the performance criteria involves the conditional expectation of a random variable given that information. Such a problem is known as Bayesian persuasion or information design which has seen a variety of applications in finance, politic science, law enforcement, medical testing and so on. Our goal is to develop the mathematical theory and numerical algorithm for both static and dynamic information design problems, possibly with information constraints, and static information games. The other line of research incorporates biological models of evolution and learning based on the principle of selection and mutation into the financial market. Both topics bring interesting problems and ideas from other fields such as economics and biology to the stochastic control and mathematical finance communities, which will open doors for new theories and applications.