Equilibrium with Randomized Strategies in Learning Theory and Mathematical Finance

该研究项目的目标是有助于理解可能具有相互矛盾的利益的代理人之间的相互作用。该项目涉及两个主题，其中代理的最优策略是通过特殊的随机化获得的。第一个主题是金融数学，涉及金融市场中的信息不对称。在这个框架中，感兴趣的主要问题是理解代理如何相互作用，以及如果一些代理具有更高或更低的信息，价格过程是如何演变的。该研究项目有望带来金融市场风险管理的新工具。第二个主题是学习理论，考虑代理(学习者)和对手之间的相互作用，前者旨在根据这些事件的额外信息预测未来事件的结果，后者旨在使学习者的任务尽可能困难。在这个带有专家建议的预测框架中，研究者的目标是为学习者得出最优的学习策略。更具体地说，对于第一个主题，研究者将建立长期不对称信息的金融市场均衡的存在性。与通过Hamilton-Jacobi-Bellman方程来描述问题的经典形式不同，研究者将通过使用凸分析和最优传输的工具来找到均衡。然后，在随机流动性和具有自然分布假设的风险厌恶代理人等问题的不同扩展中，研究均衡策略和定价规则的性质。对于第二个主题，学习者和对手之间的互动将被描述为零和随机博弈。然后，将使用偏微分方程组、随机分析和平均场理论的工具来研究这些游戏的长期行为。一个重要的目标将是找到渐近纳什均衡的简单特征，并评估经典学习算法的性能。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The objective of the research project is to contribute to the understanding of the interaction between agents that might have contradictory interests. The project addresses two topics where the optimal strategies of the agents are obtained via particular randomization. The first topic is in financial mathematics and concerns the asymmetry of information in financial markets. In this framework, the main question of interest is to understand how the agents interact and how the price processes evolve if some of the agents have superior or inferior information. The research project is expected to lead to novel tools of risk management in financial markets. The second topic is in learning theory and considers the interaction between an agent (the learner) who aims to predict the outcome of future events based on additional information on these events and an adversary who aims to make the task of the learner as difficult as possible. In this prediction with expert advice framework, the objective of the investigator is to derive optimal learning strategies for the learner. Graduate students are involved in the project.To be more specific, regarding the first topic, the investigator will establish the existence of equilibrium in financial markets with long-lived asymmetric information. Unlike the classical formulation of the problem via Hamilton-Jacobi-Bellman equations, the investigator will find an equilibrium by using tools from convex analysis and optimal transport. Then, the properties of the equilibrium strategies and pricing rules will be studied in various extensions of the problem such as stochastic liquidity and risk-averse agents with natural distributional assumptions. For the second topic, the interaction between the learner and the adversary will be stated as a zero-sum stochastic game. Then, the long-time behavior of these games will be studied using tools from partial differential equations, stochastic analysis, and mean-field theory. An important objective will be to find simple characterizations of asymptotic Nash equilibria and to assess the performances of classical learning algorithms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.