High-Dimensional Open Markets and Long-Term Investing

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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.