Stochastic Equilibria and Related Topics in Financial Mathematics

研究者研究了两个与金融市场行为有关的问题。在第一部分中，他研究了不完全市场中的均衡。市场均衡理论很重要，因为它提供了一个理解市场中价格如何发展的框架。粗略地说，如果每个代理人都能与其他代理人就每种可能的未来状态进行交易，那么市场就是完备的。完全市场的理论已经相当完善，但不完全市场的均衡是开放的。在第二部分中，他研究了在考虑如何使用随机禀赋进行最优投资时出现的随机控制问题。金融市场的形成是一个复杂的过程，但最终依赖于基本的经济原则。正是这种依赖使得它适合于数学分析和定量研究。更好地理解真正的金融市场是如何形成的，以及它们的显著特征是如何从基本构建块中产生的，这些好处是多方面的。它们的范围从提高我们监管现有和新兴市场的能力，到建立预防未来美国和全球金融不稳定和危机的工具。研究生也包括在这个项目中。研究者和他的学生研究数学金融和随机控制理论中两个独立但相关的问题群。第一部分旨在阐明理性主体对不同市场结构和不完全金融市场形成的最优反应。除了经典的凸分析和泛函分析工具外，研究者的方法还依赖于随机分析和后向随机微分方程理论，以及它们与偏微分方程系统的关系，涉及黎曼几何和非合作博弈论中的主题。第二个研究重点——围绕一类被称为“弱约束”的新型随机控制问题——是根据数学金融中的一个基本问题，即具有随机禀赋的最优投资问题来建模的。这些弱约束问题自然发生，具有有趣的数学特征，但无法使用标准方法进行分析。

The investigator studies two problems related to behavior of financial markets. In the first he studies equilibrium in incomplete markets. The theory of market equilibrium is important because it provides a framework for understanding how prices develop in a market. Roughly speaking, a market is complete if every agent can trade with every other agent about every possible future state. The theory for complete markets has been worked out reasonably well, but the equilibrium of incomplete markets is open. In the second, he examines stochastic control problems that arise in considering how to make optimal investments with a random endowment. Formation of financial markets is a complex process, but ultimately reliant on fundamental economic principles. This reliance is what makes it amenable to mathematical analysis and quantitative study. The benefits of better understanding of how real financial markets come to exist, and how their salient features emerge from the basic building blocks, are multiple. They range from enhancing our ability to regulate both existing and nascent markets to building tools for prevention of future financial instabilities and crises in the US and worldwide. Graduate students are included in the project.The investigator and his students study two separate, but related, clusters of problems in mathematical finance and stochastic-control theory. The first one aims to elucidate the rational agents' optimal response to diverse market structures and the formation of incomplete financial markets. In addition to the classical, convex- and functional-analytic tools, the investigator's approach rests on stochastic analysis and the theory of backward stochastic differential equations, as well as their relationship with systems of partial differential equations, touching upon topics in Riemannian geometry and non-cooperative game theory. The second research focus -- centered around a novel class of stochastic control problems termed "weakly constrained" -- is modeled after one of the fundamental questions in mathematical finance, namely the optimal investment problem with a random endowment. These weakly constrained problems occur naturally and possess intriguing mathematical features, but resist analysis using standard approaches.