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Risk functionals with convex level sets
Mathematical Finance ( IF 1.6 ) Pub Date : 2020-05-27 , DOI: 10.1111/mafi.12270
Ruodu Wang 1 , Yunran Wei 2
Affiliation  

We analyze the “convex level sets” (CxLS) property of risk functionals, which is a necessary condition for the notions of elicitability, identifiability, and backtestability, popular in the recent statistics and risk management literature. We put the CxLS property in the multidimensional setting, with a special focus on signed Choquet integrals, a class of risk functionals that are generally not monotone or convex. We obtain two main analytical results in dimension one and dimension two, by characterizing the CxLS property of all one‐dimensional signed Choquet integrals, and that of all two‐dimensional signed Choquet integrals with a quantile component. Using these results, we proceed to show that under some continuity assumption, a comonotonic‐additive coherent risk measure is co‐elicitable with Value‐at‐Risk if and only if it is the corresponding Expected Shortfall. The new findings generalize several results in the recent literature, and partially answer an open question on the characterization of multidimensional elicitability.

中文翻译:

具有凸水平集的风险函数

我们分析了风险功能的“凸水平集”(CxLS)属性,这是在最近的统计数据和风险管理文献中流行的可引发性,可识别性和可回测性概念的必要条件。我们将CxLS属性放在多维设置中,特别关注带符号的Choquet积分,这是一类通常不是单调或凸的风险函数。通过表征所有一维有符号Choquet积分和所有二维有符号Choquet积分具有分位数分量的CxLS性质,我们获得了第一维和第二维的两个主要分析结果。利用这些结果,我们继续证明在某些连续性假设下,当且仅当它是相应的“预期不足”时,才可以与风险单价一起提出单调加性相干风险度量。新发现概括了最近文献中的一些结果,并部分回答了关于多维可诱导性表征的未解决问题。
更新日期:2020-05-27
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