Equilibrium with Randomized Strategies in Learning Theory and Mathematical Finance

该研究项目的目的是促进对可能具有相互矛盾的利益的代理之间的相互作用的理解。该项目解决了两个主题，通过特定的随机化获得代理的最佳策略。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要了解如何代理人相互作用和价格过程如何演变，如果一些代理人有上级或劣质信息。该研究项目预计将导致金融市场风险管理的新工具。第二个主题是学习理论，并考虑代理（学习者）之间的互动，其目的是根据这些事件的额外信息预测未来事件的结果，而对手的目的是使学习者的任务尽可能困难。在专家建议预测框架中，研究者的目标是为学习者获得最佳的学习策略。本课题的研究对象是研究生，具体来说，第一个课题是在长期信息不对称的金融市场中，证明均衡的存在性。与通过Hamilton-Jacobi-Bellman方程的问题的经典公式不同，研究人员将使用凸分析和最优运输的工具找到平衡。然后，均衡策略和定价规则的性质将在各种扩展的问题，如随机流动性和风险厌恶代理与自然分布的假设进行研究。对于第二个主题，学习者和对手之间的互动将被视为一个零和随机博弈。然后，这些游戏的长期行为将使用偏微分方程，随机分析和平均场理论的工具进行研究。一个重要的目标将是找到简单的渐近纳什均衡的特征，并评估经典的学习algorithm.This奖项反映了NSF的法定使命的表现，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.