Optimal Transport of Stochastic Processes in Mathematical Finance

该项目研究随机过程的最优运输，即如何以某种成本最优的方式将概率度量(即随机过程的定律)联系起来。随着计算的进步，最优传输理论已经成为在非参数高维统计、图像识别、机器学习和数学金融中的校准问题等广泛应用中不可或缺的工具。该项目将促进随机过程最优运输理论的研究。一个特别的重点将放在数学金融的应用上，例如最优的运输和时间动态效用优化问题。该理论将为数值例程和新的模型不确定性指标的发展奠定基础，这将有助于决策者为最坏情况做好准备。本项目特别关注最优运输的自适应Wasserstein距离和熵正则化，这是计算高维最优运输问题的首选方法。该项目的第一部分研究依赖于时间的稳健优化问题的一个凸对偶结果和一个一阶逼近结果，以及它们的最坏情况优化器的特征。然后，这些结果被应用于量化稳健投资组合优化和Davis定价中的模型不确定性，以及金融市场中时间依赖分布的机器学习和对冲。对于这些问题中的每一个，都导出了封闭形式的表达式，并用数值实现了这些表达式。该项目的第二部分通过一个强紧致性结果和近似技术来研究薛定谔势的稳定性--熵最优传输问题的对偶优化器。此外，基于参考模型和由市场价格得到的边际分布，以及具有熵惩罚的鞅最优运输问题的最优者的特征，导出了具有有限熵的校准鞅措施存在的充要条件。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project investigates optimal transportation of stochastic processes, that is, how to relate probability measures (i.e., the laws of stochastic processes) in a certain cost-optimal way. Amplified by computational advances, optimal transport theory has become an indispensable tool for far-reaching applications in non-parametric high dimensional statistics, image recognition, machine learning and calibration problems in mathematical finance. The project will advance research in the theory of optimal transport of stochastic processes. A special emphasis will be placed on applications to mathematical finance, such as martingale optimal transport and time-dynamic utility optimization problems. The derived theory will lay the basis for advances of numerical routines and novel indicators of model uncertainty, which will help to prepare decision-makers for worst-case scenarios.This project especially focuses on the adapted Wasserstein distance and entropic regularization of optimal transport, which is the method of choice for computing optimal transport problems in high dimensions. The first part of the project investigates a convex duality result and a first-order approximation result for time-dependent robust optimization problems and a characterization of their worst-case optimizers. These results are then applied to quantify model uncertainty in robust portfolio optimization and Davis pricing, machine learning of time-dependent distributions and hedging in financial markets. For each of these problems, closed-form expressions are derived and these are implemented numerically. The second part of the project investigates stability of Schroedinger potentials -- the dual optimizers of the entropic optimal transport problem -- via a strong compactness result and approximation techniques. Furthermore, necessary and sufficient conditions for existence of calibrated martingale measures with finite entropy are derived on the basis of a reference model and marginal distributions derived from market prices, as well as a characterization of the optimizers of the martingale optimal transport problem with entropic penalization.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.