Optimal Transport, Interacting Particles, and Stochastic Portfolio Theory

本研究课题所研究问题的现实动因来自于定量金融学。具体地说，该项目研究的是，在波动性存在的情况下，表现优于S等标准指数的投资组合背后的数学原理。这种投资组合的效用在于，在这一过程中，它们明显降低了波动性，从而有助于金融市场的稳定。尽管这种投资组合的临时使用很普遍，但到目前为止，系统的数学研究还很有限。事实证明，数学与概率和几何中的一些非常现代的主题有关，特别是Monge-Kantorovich最优运输地图的数学。成功完成这项研究不仅将在概率和信息几何方面带来惊人的数学成就，还将意味着在现代投资组合管理中具有非常实际的应用，通过提高金融稳定性，潜在的社会效益将达到最高水平。这位研究人员计划研究一系列问题，将最优传输和相互作用的粒子系统领域与现代投资组合理论的应用联系起来。这个项目包括两个部分。第一章研究指数随机积分的性质，其中被积函数由Monge-Kantorovich最优传输映射的解给出。这项研究是单位单纯形的几何和随机积分的普遍行为之间有趣的相互作用，当它们不是鞅时。经典概率在很大程度上依赖于随机积分的鞅性质。在这里，调查者探索了一类完全不同的行为，这些行为应该是独立感兴趣的。第二部分研究了有限维和无限维相互作用的连续时间粒子系统的性质。这些粒子系统起源于随机投资组合理论，可以被认为是整数格上离散时间排除过程的连续时间模拟。然而，与排斥过程不同的是，人们对这些过程知之甚少。推测它们会显示出惊人的相变。该项目旨在为这些过程建立浓度不等和波动估计(以及其他性质)，这是朝着证明这些过程更微妙的行为迈出的一步。

The practical motivation for the questions under study in this research project comes from quantitative finance. Specifically, the project investigates the mathematics behind portfolios that outperform a standard index such as S&P 500 in the presence of volatility. The utility of such portfolios is that in that process they reduce volatility demonstrably, and thus contribute to the stability of financial markets. Although ad hoc use of such portfolios is widespread, a systematic mathematical study has been limited so far. It turns out the mathematics is related to some very modern topics in probability and geometry, especially that of the Monge-Kantorovich optimal transport maps. Successful completion of the research will not only lead to striking new mathematics in probability and information geometry, but will also imply very practical applications in modern portfolio management, with potential societal benefits of the highest level via increased financial stability. The investigator plans to study an array of problems linking the fields of optimal transport and interacting particle systems, with applications to modern portfolio theory. There are two parts to this project. The first one investigates the behavior of exponential stochastic integrals where the integrand is given by a solution of a Monge-Kantorovich optimal transport map. The study is an interesting interplay between geometry of the unit simplex and universal behavior of stochastic integrals when they are not martingales. Classical probability relies heavily on the martingale property of stochastic integrals. Here, the investigator explores a completely distinct class of behaviors that should be of independent interest. The second part studies properties of interacting continuous-time particle systems in finite and infinite dimensions. These particle systems, originally coming from stochastic portfolio theory, can be thought of as a continuous-time analogue of the discrete time exclusion process on the integer lattice. However, unlike exclusion processes, very little is known about these processes. They are conjectured to display striking phase transitions. The project aims to establish concentration inequalities and fluctuation estimates (among other properties) for such processes, which is a step towards proving more subtle behavior of these processes.