High-Dimensional Open Markets and Long-Term Investing

对大型股票市场的长期投资对包括养老基金、共同基金和大学捐赠基金在内的广泛投资者来说是至关重要的。这项研究通过在随机投资组合理论的数学框架内加深对这类投资的理解来支持他们的决策。该项目促进了数学金融与相互作用粒子系统、平均场分析和极值统计等领域之间的思想、技术和工具的转移。通过这个项目组织的活动使博士后、研究生和本科生的不同群体受益。研究重点是通过分析公开市场的显式、可处理的模型来对高维市场进行长期投资。公开市场是指可用资产的集合随着时间的推移而变化。最近开发的混合和秩雅可比模型占据了中心舞台。这些模型可以被视为基于等级的相互作用的粒子系统，其值在标准单纯形中，并且非常容易处理和服从详细的分析。这项研究涉及一些相互依赖的问题，包括参数估计、典型粒子动力学的平均场分析、无限大的市场、资本分布曲线的上缘渐近性和上阶统计量的混沌传播。与不太具体的框架相比，易处理的设置允许更详细的理解。除了具有内在的兴趣之外，通过这种分析产生的数学和财务见解可以通过模型不确定性和稳健性分析转移到更一般的情况。该奖项反映了NSF的法定使命，并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。

Long-term investments in large equity markets are of fundamental importance to a wide range of investors including pension funds, mutual funds, and university endowments. This research supports their decision making by deepening the understanding of such investments within the mathematical framework of Stochastic Portfolio Theory. The project advances the transfer of ideas, techniques, and tools between mathematical finance and areas such as interacting particle systems, mean-field analysis, and extreme value statistics. A diverse group of postdocs, graduate, and undergraduate students benefits from the activities organized through this project.The research focuses on long-term investing in high-dimensional markets through the analysis of explicit, tractable models of open markets. An open market is one where the set of available assets varies over time. The recently developed hybrid and rank Jacobi models take center stage. These models can be viewed as rank-based interacting particle systems with values in the standard simplex, and are extremely tractable and amenable to detailed analysis. The research is concerned with a number of interdependent problems, including parameter estimation, mean-field analysis of representative particle dynamics, markets of infinite size, upper edge asymptotics of the capital distribution curve, and propagation of chaos for the upper order statistics. The tractable setup allows for a more detailed understanding than that afforded by less specific frameworks. In addition to being of intrinsic interest, the mathematical and financial insights generated through this analysis are transferable to more general situations by means of a model uncertainty and robustness analysis.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.