The main aim of this paper is to introduce the notion of risk excess measure, to analyze its properties, and to describe some basic construction methods. To compare the risk excess of one distribution Q w.r.t. a given risk distribution P, we apply the concept of hemi-metrics on the space of probability measures. This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures. Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings. Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the values ρ(Q), ρ(P) of a risk measure ρ. We argue that the difference ρ(Q)−ρ(P) neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure. We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.

中文翻译：

---

半测量法诱发的风险超额措施

本文的主要目的是介绍风险超额计量的概念，分析其性质，并描述一些基本的构建方法。为了比较给定风险分布P与一个分布Q的风险超额，我们将半度量的概念应用于概率测度的空间。这种风险比较的观点在将基础空间的排序和半度量扩展到概率测度水平方面具有自然的基础。这类扩展的基本示例是由大众运输和功能类诱导的排序引起的。我们对风险超额度量的观点增加了通常考虑的方法，即通过风险度量ρ的值ρ（Q），ρ（P）比较Q和P的风险。我们认为，差异ρ（Q）-ρ（P）忽略了风险超额的相关方面，这些方面已由新的风险超额度量概念充分描述。我们得出了各种风险超额措施的具体类别，并讨论了相应的排序和措施扩展属性。