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测度（即随机过程的规律）联系起来。随着计算技术的进步，最优运输理论已经成为非参数高维统计、图像识别、机器学习和数学金融校准问题等广泛应用中不可或缺的工具。该项目将推进随机过程最优输运理论的研究。特别强调数学金融的应用，如鞅最优运输和时间动态效用优化问题。该理论将为数值程序的发展和模型不确定性的新指标奠定基础，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输运的自适应Wasserstein距离和熵正则化，这是计算高维最优输运问题的首选方法。项目的第一部分研究了时变鲁棒优化问题的凸对偶结果和一阶近似结果，以及它们的最坏情况优化器的表征。然后将这些结果应用于量化稳健投资组合优化和戴维斯定价中的模型不确定性，时间相关分布的机器学习以及金融市场中的对冲。对于每一个问题，都推导出了封闭形式的表达式，并用数值方法实现了这些表达式。项目的第二部分通过强紧性结果和近似技术研究了薛定谔势的稳定性——熵最优输运问题的对偶优化器。此外，基于参考模型和由市场价格导出的边际分布，给出了有限熵鞅最优运输问题存在的充分必要条件，并刻画了具有熵惩罚的鞅最优运输问题的优化器。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。

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.