Systems of Backward Stochastic Differential Equations and Applications in Stochastic Financial Equilibrium Theory

为什么股票价格图表看起来是这样的？为什么会出现市场崩溃和反弹？资产价格在多大程度上取决于我们的风险偏好，又在多大程度上取决于我们周围世界的内在随机性？这些问题可以通过分析各种数学模型来回答，这些模型都是基于价格调整以使供需平衡的基本思想。这种分析通常是非常困难的，需要相当大的分析努力，但仍然是当代数学及其工具所能达到的。更好地理解资产价格变动背后的结构，其社会效益数不胜数；例如，它将帮助我们更有效地监管市场，更好地规划未来潜在的金融不稳定。首席研究员和他的学生和合作者将在概率、数学金融和随机控制理论的交叉点研究几个独立但最终相关的问题群。一个集中的领域是被称为倒向随机微分方程（BSDEs）的一类方程，它描述了许多不同情况下随机过程的进化。PI和他的合作者将专注于非线性全耦合BSDEs系统的存在唯一性理论。虽然他们对这门学科的兴趣源于它在金融均衡理论中的关键作用，但在金融、经济学、博弈论、最优随机控制和其他领域还有许多其他应用。第二组问题围绕均衡理论中所谓的平均场极限。这里的主要目标是对市场由大量小型经济主体组成时的均衡问题提供有意义的简化。研究了指数投资者不完全市场均衡模型中BSDEs系统的极限行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Why do stock-price graphs look the way they do? Why do market crashes and rallies occur? How much do asset prices depend on our risk appetites, and how much on the inherent randomness of the world around us? These questions can be answered by analyzing various mathematical models based on the fundamental idea that prices adjust to equalize supply with demand. This analysis is often very hard, and requires considerable analytical effort, but is nevertheless within the reach of contemporary mathematics and its tools. The societal benefits of a better understanding of the structure behind asset-price movements are numerous; for instance, it will help us regulate the markets more efficiently and plan better for potential future financial instabilities. The principal investigator and his students and collaborators will study several separate, but ultimately related, clusters of problems at the intersection of probability, mathematical finance and stochastic-control theory. One area of concentration is the class of equations, known as Backward Stochastic Differential Equations (BSDEs), which describe evolutions of random processes in many different contexts. The PI and his collaborators will focus on the existence and uniqueness theory for systems of nonlinear fully-coupled BSDEs. While their interest in this subject originates in its pivotal role in the financial equilibrium theory, there are numerous other applications in finance, economics, game theory, optimal stochastic control, and other fields. The second cluster of problems centers around the so-called mean-field limits in the context of equilibrium theory. The main goal here is to provide a meaningful simplification of the equilibrium problem when the market consists of a large number of small economic agents. Attention will be paid to the limiting behavior of the systems of BSDEs which appear in incomplete-market equilibrium models with exponential investors.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.