Distorted optimal transport

TL;DR

This work introduces distorted optimal transport by minimizing a distorted expectation $\mathcal{E}^h[c(X,Y)]$ under marginal constraints, linking probability, decision theory, and risk management. On the real line, the authors show that when $h$ is convex and the cost $c$ is submodular and monotone, the comonotonic coupling $C^+$ is universally optimal, and they establish weak duality with strong duality for affine costs. For linear costs, they obtain uniqueness of universal optimizers under strictly convex or strictly concave distortions; for inverse S-shaped distortions and S-shaped distortions, they characterize universal optimizers via novel copula constructions ($C_p^ frac{+}{-}$ and $C_p^ p$) and a sharp condition $h'(p^+)\,\ge\,h'(0^+)$, revealing a rich, tail-region dependent optimal structure. The framework also yields results for risk measures and risk aggregation, e.g., RVaR and VaR scenarios, and includes an applied economic production example illustrating the practical implications. Overall, the paper provides new mathematical links between distorted probability, optimal transport, and risk theory, with multiple open questions for extensions and stronger duality results.

Abstract

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 non-linear expectation. This new formulation is motivated by concrete problems in decision theory, robust optimization, and risk management, and it has many distinct features compared to 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 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 counter-monotonic" dependence structure; for S-shaped 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.

Figures (7)

• Figure 1: The transport maps between two normal distributions described by the copulas $C^+$ (left) and $C^-$ (right)
• Figure 2: ISS distortion functions that are strictly ISS (left, with a unique inflection point) and not strictly ISS (right, with many inflection points)
• Figure 3: Supports of the copulas $C_{p}^\pm$ (left) and $C_p^\mp$ (right)
• Figure 4: The transport maps between two normal distributions described by the copulas $C^{\pm}_p$ (left) and $C^{\mp}_p$ (right) with $p=0.5$
• Figure 5: Supports of the copulas $C^\pm_{0.5}$ (left) and $C^*$ (right), where the numbers represent the corresponding weight of each line segment

Theorems & Definitions (33)

• Definition 1: Universal optimality
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• proof
• Theorem 2
• proof
• Example 1
• Lemma 4