凸损失下的两区制风险度量
Two-Regime Risk Measures under Convex Loss
- University of Bucharest（布加勒斯特大学）
- Institute of Mathematics of the Romanian Academy（罗马尼亚科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出凸损失下两区制风险度量，通过最小化残差损失确定边界与代表水平，证明其收敛性并分析公理失效，给出算法与示例。
AI中文摘要：
我们研究实值损失分布的两区制汇总。两个代表性水平及其边界通过最小化凸残差损失来选择。当分布在边界处存在原子时，将该原子分配给下区制或上区制可能产生不同的优化成本。采用更好的整体原子分配可得到下半连续的截断轮廓。我们证明了该轮廓在损失自适应的ψ-弱拓扑下的epi-收敛性，并获得了最小值收敛性及最优截断集的外稳定性。在唯一的有限无原子极限截断、正区制质量和唯一条件中心下，两个拟合水平及规范的单跳表示也收敛。我们还从货币风险公理的角度审视所得汇总，建立了外部单调性和次可加性的明确失效，并给出了具有条件误差控制的有限支撑算法。一个四情景模型说明了截断几何以及平局和原子的作用。
英文摘要：
We study a two-regime summary of a real-valued loss distribution. The two representative levels and the boundary between them are chosen by minimizing a convex residual loss. When the distribution has an atom at the boundary, assigning that atom to the lower or upper regime can give different optimized costs. Taking the better whole-atom assignment yields the lower-semicontinuous cutoff profile. We prove epi-convergence of this profile under a loss-adapted \(ψ\)-weak topology and obtain convergence of minimum values and outer stability of optimal cutoff sets. Under a unique finite atom-free limiting cutoff, positive regime masses, and unique conditional centers, the two fitted levels and a canonical one-jump representation converge as well. We also examine the resulting summary from the viewpoint of monetary risk axioms, establish explicit failures of external monotonicity and subadditivity, and give finite-support algorithms with conditional error control. A four-scenario model illustrates the cutoff geometry and the role of ties and atoms.
