Multivariate risks and depth-trimmed regions

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• 作者：Ignacio Cascos and Ilya Molchanov
• 关键词：Acceptance set ; Cone ; Depth ; trimmed region ; Multivariate risk ; Risk measure
• 刊名：Finance and Stochastics
• 出版年：2007
• 出版时间：July, 2007
• 年：2007
• 卷：11
• 期：3
• 页码：373-397
• 全文大小：533 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Quantitative Finance
  Finance and Banking
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Economic Theory
  Probability Theory and Stochastic Processes
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1122

文摘

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.