Stochastic Equilibria and Related Topics in Financial Mathematics

调查者研究了两个与金融市场行为相关的问题。首先，他研究了不完全市场中的均衡。市场均衡理论很重要，因为它为理解市场价格如何发展提供了一个框架。粗略地说，如果每个代理人都能就每一个可能的未来状态与其他代理人进行交易，那么一个市场就是完整的。关于完全市场的理论已经被提出得相当好，但不完全市场的均衡是开放的。在第二章中，他研究了在考虑如何利用随机天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，但最终取决于基本的经济原则。正是这种依赖使它易于进行数学分析和定量研究。更好地了解真正的金融市场是如何存在的，以及它们的显著特征是如何从基本构件中显现出来的，这些好处是多方面的。这些措施包括加强我们监管现有和新兴市场的能力，以及建立工具以防止美国和全球未来的金融不稳定和危机。研究人员和他的学生研究数学金融和随机控制理论中两个独立但又相互关联的问题。第一章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、凸的和泛函分析工具外，研究人员的方法还依赖于随机分析和倒向随机微分方程组的理论，以及它们与偏微分方程组的关系，涉及黎曼几何和非合作博弈论的主题。第二个研究重点--围绕着一类新的随机控制问题，称为“弱约束”--仿照数学金融学中的一个基本问题，即具有随机禀赋的最优投资问题。这些弱约束问题是自然产生的，具有有趣的数学特征，但拒绝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.