Distorted Optimal Transport

• 成果类型: Article; Early Access
• 署名作者: Liu, Haiyan; Wang, Bin; Wang, Ruodu; Zhuang, Sheng Chao
• 署名单位: Michigan State University; Michigan State University; Chinese Academy of Sciences; University of Waterloo; University of Nebraska System; University of Nebraska Lincoln
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2024.0591
• 发表日期: 2025
• 关键词: model uncertainty; expected utility; risk aggregation; probability; REPRESENTATION; Robustness

摘要

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.