Distorted Optimal Transport
提出扭曲最优传输框架，通过最小化非线性期望下的扭曲期望成本来扩展经典最优传输理论，并研究不同扭曲函数对最优传输计划的影响，为概率、决策理论和风险管理提供新数学桥梁。
Classic optimal transport theory is formulated through minimizing the expected transport cost between two given distributions. We propose the framework of distorted optimal transport by minimizing a distorted expected cost, which is the cost under a nonlinear expectation. This new formulation is motivated by concrete problems in decision theory, robust optimization, and risk management, and it has many distinct features compared with the classic theory. We choose simple cost functions and study different distortion functions and their implications on the optimal transport plan. We show that on the real line, the comonotonic coupling is optimal for the distorted optimal transport problem when the distortion function is convex and the cost function is submodular and monotone. Some forms of duality and uniqueness results are provided. For inverse-S-shaped (SS) distortion functions and linear cost, we obtain the unique form of optimal coupling for all marginal distributions, which turns out to have an interesting “first comonotonic, then countermonotonic” dependence structure; for SS distortion functions, a similar structure is obtained. Our results highlight several challenges and features in distorted optimal transport, offering a new mathematical bridge between the fields of probability, decision theory, and risk management. Funding: R. Wang is supported by the Natural Sciences and Engineering Research Council of Canada [Grants CRC-2022-00141 and RGPIN-2024-03728].