{"title":"An elementary proof of the dual representation of Expected Shortfall","authors":"Martin Herdegen, Cosimo Munari","doi":"10.1007/s11579-023-00346-8","DOIUrl":null,"url":null,"abstract":"<p>We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall.</p>","PeriodicalId":48722,"journal":{"name":"Mathematics and Financial Economics","volume":"39 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics and Financial Economics","FirstCategoryId":"96","ListUrlMain":"https://doi.org/10.1007/s11579-023-00346-8","RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"BUSINESS, FINANCE","Score":null,"Total":0}
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Abstract
We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall.
期刊介绍：
The primary objective of the journal is to provide a forum for work in finance which expresses economic ideas using formal mathematical reasoning. The work should have real economic content and the mathematical reasoning should be new and correct.
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