Optimal Transport of Stochastic Processes in Mathematical Finance

该项目研究随机过程的最佳运输，即如何将概率测度（即，随机过程的规律）以某种成本最优的方式。随着计算技术的进步，最优传输理论已成为非参数高维统计、图像识别、机器学习和数学金融中校准问题等广泛应用的不可或缺的工具。该项目将推进随机过程最佳运输理论的研究。一个特别的重点将放在应用数学金融，如鞅最优运输和时间动态效用优化问题。推导出的理论将奠定基础的数值例程和模型的不确定性的新指标，这将有助于决策者准备的最坏情况scenaries.This项目的进步，特别是适应Wasserstein距离和熵正则化的最佳运输，这是计算最佳运输问题的方法选择在高维。该项目的第一部分研究了时间相关鲁棒优化问题的凸对偶结果和一阶逼近结果以及其最坏情况优化器的特征。然后，这些结果被应用到量化模型的不确定性鲁棒投资组合优化和戴维斯定价，机器学习的时间依赖分布和对冲在金融市场。对于这些问题中的每一个，推导出封闭形式的表达式，这些数值实现。该项目的第二部分研究薛定谔电位的稳定性-熵最优运输问题的对偶优化器-通过强紧性结果和近似技术。进一步，基于参考模型和市场价格的边际分布，得到了有限熵标定鞅测度存在的充要条件，该奖项反映了美国国家科学基金会的法定使命，并被认为是值得支持的，通过评估使用基金会的知识产权，优点和更广泛的影响审查标准。

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.