This study develops closed-form solutions for distortion risk measures (DRM) in extreme cases by utilizing the first two moments and the symmetry of underlying distributions. The resultant extreme-case distributions, encompassing the worst- and best-case distributions, are identified by the envelopes of the distortion functions. The findings of this study extend previous research on worst-case risk measures such as worst-case VaR, worst-case CVaR, worst-case RVaR, and worst-case spectral risk measure, by presenting a unified framework. Furthermore, the compact solutions enhance tractability in optimization problems involving these risk measures, particularly when the true underlying distribution is unknown, and the first two moments are uncertain. The application of the extreme-case DRMs is illustrated with real data sets through numerical examples.

本研究利用前两个矩和基础分布的对称性，为极端情况下的失真风险度量（DRM）开发了封闭式解决方案。由此产生的极端情况分布（包括最坏情况和最佳情况分布）由畸变函数的包络线来识别。本研究的结果通过提出一个统一的框架，扩展了先前对最坏情况风险度量（例如最坏情况 VaR、最坏情况 CVaR、最坏情况 RVaR 和最坏情况频谱风险度量）的研究。此外，紧凑的解决方案增强了涉及这些风险度量的优化问题的易处理性，特别是当真实的基础分布未知且前两个时刻不确定时。