Optimal Transport, Interacting Particles, and Stochastic Portfolio Theory

最优传输、相互作用粒子和随机投资组合理论

• 批准号: 1612483
• 负责人: Soumik Pal
• 金额: $ 18.04万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612483&HistoricalAwards=false
• 关键词: Optimal; Transport; Interacting; Particles; Stochastic

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.

本研究课题所研究问题的现实动机来自于量化金融。具体来说，该项目研究了在波动性存在的情况下，投资组合表现优于标准指数（如标准普尔500指数）背后的数学。这种投资组合的效用是，在这一过程中，它们明显减少了波动，从而有助于金融市场的稳定。虽然特设使用这种投资组合是广泛的，系统的数学研究一直有限。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最优运输映射。研究的成功完成不仅将导致概率和信息几何学中引人注目的新数学，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。研究人员计划研究一系列问题，将最佳运输和相互作用的粒子系统领域联系起来，并应用于现代投资组合理论。这个项目有两个部分。第一个研究指数随机积分的行为，其中被积函数是由Monge-Kantorovich最优迁移映射的解给出的。这项研究是一个有趣的相互作用的几何单位单纯形和普遍行为的随机积分时，他们不是鞅。经典概率在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。第二部分研究有限维和无限维相互作用连续时间粒子系统的性质。这些粒子系统，最初来自随机投资组合理论，可以被认为是一个连续时间模拟的离散时间排斥过程的整数格。然而，与排斥过程不同，人们对这些过程知之甚少。它们被用来显示惊人的相变。该项目旨在为这些过程建立浓度不等式和波动估计（以及其他属性），这是朝着证明这些过程的更微妙行为迈出的一步。