Entropy in Optimal Transport and Finance

本项目研究了两种情况下的熵罚：最优运输和金融。在最优传输中，熵正则化是一种近似值，可以在机器学习或图像处理等数据丰富的环境中实现快速而稳健的计算。在金融领域，熵罚提供了一种灵活而系统的方法，可以根据期权价格的市场数据校准基准证券模型。这项研究推动了科技和科学领域的广泛应用，在这些领域，熵的方法被使用，并导致了知识在最优运输、数学金融和概率论之间的转移。作为项目的一部分，培训了一批不同的博士后、研究生和本科生。本课题的第一部分研究了熵正则化最优传输（EOT）。最优传输提供了一种自然的方式，将距离或成本从基本空间提升到其概率度量空间，因此在比较数据集或统计分布的应用中无处不在。熵正则化允许Sinkhorn算法，是高维环境下最重要的近似计算方法。该项目开发了一种新的几何方法来研究随着正则化参数的减小EOT对非正则化对偶的收敛性。此外，它研究了EOT相对于边际分布的稳定性，以及解决相关对偶问题的所谓Schrödinger势。本文的第二部分研究了金融中期权定价模型的校准问题。从参考模型（如标准随机波动率模型）和市场给出的期权价格数据出发，在拟合数据的所有鞅中，通过最小化相对于参考的相对熵来选择校准模型。这种非参数方法是数据驱动和灵活的，同时保留了参考模型的理想品质。校准模型是一个模拟经典的Schrödinger桥接过程，但包含一个额外的鞅约束，以避免动态套利。该项目从金融和概率的角度来研究这个模型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates entropic penalties in two contexts: optimal transport and finance. In optimal transport, entropic regularization is an approximation enabling fast and robust computations in data-rich settings such as machine learning or image processing. In finance, entropic penalties yield a flexible and systematic method to calibrate a benchmark security model to market data on option prices. The research advances a vast array of applications in technology and science where entropic methods are used and leads to a transfer of knowledge between optimal transport, mathematical finance and probability theory. A diverse group of postdocs, graduate and undergraduate students is trained as part of the project.The first part of this project investigates entropically regularized optimal transport (EOT). Optimal transport provides a natural way to lift a distance or cost from a base space to its space of probability measures, hence has become ubiquitous in applications where data sets or statistical distributions are compared. Entropic regularization allows for Sinkhorn's algorithm and is the most important method for approximate computation is high-dimensional settings. The project develops a novel geometric method to study the convergence of EOT to its unregularized counterpart as the regularization parameter decreases. Moreover, it studies the stability of EOT with respect to the marginal distributions, as well as the so-called Schrödinger potentials that solve an associated dual problem. The second part of this proposal studies the calibration of option pricing models in finance. Starting with a reference model, such as a standard stochastic volatility model, and option price data given by the market, a calibrated model is chosen by minimizing the relative entropy with respect to the reference among all martingales fitting the data. This nonparametric approach is data-driven and flexible while retaining desirable qualities of the reference model. The calibrated model is an analogue to the classical Schrödinger bridge process but incorporates an additional martingale constraint to avoid dynamic arbitrages. The project investigates this model from financial and probabilistic perspectives.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.