标题： 引出与法律不变泛函相关的参考度量 显示英文标题

标题： Eliciting reference measures of law-invariant functionals

摘要： 规律不变泛函是风险管理的核心，它在无原子参考概率测度下，为具有相同分布的随机前景赋予相同的值。该测度通常被假设为固定不变的。在这里，我们采用相反的视角：仅给定观测到的泛函值，我们的目标是要么恢复参考测度，要么在不先验满足该性质时，确定一个候选测度来测试规律不变性。我们的方法基于对在规律不变域上定义的规律不变泛函的一个关键观察。这些泛函在有符号测度的对偶空间中定义了下（上）支撑集，并且这些支撑集的上确界（如果存在）是参考测度的标量倍数。在特定情况下，这一观察可以表述为三明治定理。我们通过对一些著名案例的详细分析来说明该方法：熵风险测度、预期损失和风险价值。对于后者，我们的引出程序最初由于支持集极值的简单性而失败。因此，我们开发了一个合适的修改方案。 显示英文摘要

摘要： Law-invariant functionals are central to risk management and assign identical values to random prospects sharing the same distribution under an atomless reference probability measure. This measure is typically assumed fixed. Here, we adopt the reverse perspective: given only observed functional values, we aim to either recover the reference measure or identify a candidate measure to test for law invariance when that property is not {\em a priori} satisfied. Our approach is based on a key observation about law-invariant functionals defined on law-invariant domains. These functionals define lower (upper) supporting sets in dual spaces of signed measures, and the suprema (infima) of these supporting sets -- if existent -- are scalar multiples of the reference measure. In specific cases, this observation can be formulated as a sandwich theorem. We illustrate the methodology through a detailed analysis of prominent examples: the entropic risk measure, Expected Shortfall, and Value-at-Risk. For the latter, our elicitation procedure initially fails due to the triviality of supporting set extrema. We therefore develop a suitable modification.