Optimal Transport, Interacting Particles, and Stochastic Portfolio Theory

本研究项目所研究问题的实际动机来自于量化金融。具体来说，该项目调查了在波动存在的情况下优于标准指数（如标准普尔500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。虽然这种组合的特别使用很普遍，但迄今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话题有关，尤其是蒙格-坎托洛维奇最优运输图。该研究的成功完成不仅会在概率论和信息几何中产生惊人的新数学，而且还将意味着在现代投资组合管理中的非常实际的应用，通过增加金融稳定性，具有最高水平的潜在社会效益。研究者计划研究一系列问题，将最佳输运和相互作用粒子系统领域与现代投资组合理论的应用联系起来。这个项目有两个部分。第一部分研究指数随机积分的性质，其中被积函数由Monge-Kantorovich最优运输图的解给出。研究了单位单纯形的几何性质与非鞅随机积分的普遍行为之间的有趣相互作用。经典概率论很大程度上依赖于随机积分的鞅性质。在这里，研究者探索了一个完全不同的行为类别，应该是独立的兴趣。第二部分研究了有限维和无限维连续时间相互作用粒子系统的性质。这些粒子系统最初来自随机组合理论，可以被认为是整数晶格上离散时间不相容过程的连续时间模拟。然而，与排斥过程不同的是，我们对这些过程知之甚少。据推测，它们显示出惊人的相变。该项目旨在为这些过程建立浓度不平等和波动估计（以及其他属性），这是证明这些过程更微妙行为的一步。

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.