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

后向随机微分方程组及其在随机金融均衡理论中的应用

基本信息

批准号：
1815017
负责人：
Gordan Zitkovic
金额：
$ 37.66万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1815017&HistoricalAwards=false
关键词：
Systems Backward Stochastic Differential Equations

项目摘要

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.

为什么股价图表会是这样的呢？为什么会出现市场崩盘和反弹？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？这些问题可以通过分析各种数学模型来回答，这些模型基于价格调整的基本思想，即供需平衡。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有很多社会好处；例如，它将帮助我们更有效地监管市场，并更好地为未来潜在的金融不稳定制定计划。主要研究人员和他的学生及合作者将研究概率、数学金融和随机控制理论交集的几个独立但最终相关的问题群。一个集中的领域是一类方程，称为倒向随机微分方程(BSDE)，它描述了随机过程在许多不同环境中的演变。PI和他的合作者将专注于非线性全耦合BSDE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，但它在金融、经济学、博弈论、最优随机控制等领域还有许多其他应用。在均衡理论的背景下，第二组问题围绕着所谓的平均场极限。这里的主要目标是，当市场由大量小型经济主体组成时，对均衡问题提供一个有意义的简化。这一奖项反映了NSF的法定使命，通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。