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？为什么市场会出现崩盘和反弹？资产价格在多大程度上取决于我们的风险偏好，在多大程度上取决于我们周围世界固有的随机性？这些问题可以通过基于价格调整以平衡供需这一基本思想的各种数学模型来回答。这种分析通常非常困难，需要大量的分析工作，但仍然在当代数学及其工具的能力范围内。更好地了解资产价格变动背后的结构可以带来很多社会效益。例如，它将帮助我们更有效地监管市场并更好地规划未来潜在的金融不稳定。首席研究员及其学生和合作者将研究概率、数学金融和随机控制理论交叉领域的几个独立但最终相关的问题群。 其中一个重点领域是一类方程，称为后向随机微分方程 (BSDE)，它描述了许多不同背景下随机过程的演化。 PI 和他的合作者将重点研究非线性全耦合倒向随机微分方程系统的存在唯一性理论。虽然他们对这一主题的兴趣源于其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。第二组问题集中在平衡理论背景下的所谓平均场极限。这里的主要目标是当市场由大量小型经济主体组成时，对均衡问题进行有意义的简化。我们将关注出现在指数投资者的不完全市场均衡模型中的 BSDE 系统的限制行为。该奖项反映了 NSF 的法定使命，并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，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.