发表机构
University of Calgary; Southern University of Science and Technology; University of Science and Technology of China（卡尔加里大学; 南方科技大学; 中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文刻画了Wasserstein球上均值-方差泛函的极值，化简为标量方程并构造位置-尺度变换，证明同质信念下比例分配最优，并扩展至扭曲-方差泛函。
AI 中文摘要
我们刻画了在 2-Wasserstein 球上均值-方差泛函的最坏情形和最好情形的值。利用分位数表示以及可达均值与标准差的几何性质，我们将这两个无穷维问题化简为标量方程，并将极值分布构造为参考分布的位置-尺度变换。它们的值仅通过参考分布的前两阶矩依赖于该分布。随后，我们推导了对偶表示，并将异质信念下的比例风险分担表述为一个有限维优化问题。在同质信念下，我们证明经典的比例分配对任意模糊半径和 α-最大最小权重仍然是最优的。最后，我们研究了一个耦合的扭曲-方差泛函，并通过凸包络构造刻画了其最坏情形分位数，使得极值分布不仅可以在位置和尺度上变化，还可以在形状上变化。
英文摘要
We characterize the worst- and best-case values of a mean-variance functional over a 2-Wasserstein ball. Using quantile representations and the geometry of attainable means and standard deviations, we reduce both infinite-dimensional problems to scalar equations and construct the extremal laws as location-scale transformations of the reference distribution. Their values depend on the reference law only through its first two moments. We then derive dual representations and formulate proportional risk sharing under heterogeneous beliefs as a finite-dimensional optimization problem. Under homogeneous beliefs, we show that the classical proportional allocation remains optimal for every ambiguity radius and $α$-maxmin weight. Finally, we study a coupled distortion-variance functional and characterize its worst-case quantile through a convex-envelope construction, allowing the extremal law to change in shape as well as location and scale.
Comments36 pages, 3 figures
