Subdifferential representations of risk measures
成果类型：
Article
署名作者：
Pflug, Georg Ch.
署名单位：
University of Vienna
刊物名称：
MATHEMATICAL PROGRAMMING
ISSN/ISSBN：
0025-5610
DOI：
10.1007/s10107-006-0714-8
发表日期：
2006
页码：
339-354
关键词：
摘要：
Measures of risk appear in two categories: Risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures (safety measures). Pure risk measures (risk deviation measures) are natural generalizations of the standard deviation. While pure risk measures are typically convex, acceptability measures are typically concave. In both cases, the convexity (concavity) implies under mild conditions the existence of subgradients (supergradients). The present paper investigates the relation between the subgradient (supergradient) representation and the properties of the corresponding risk measures. In particular, we show how monotonicity properties are reflected by the subgradient representation. Once the subgradient (supergradient) representation has been established, it is extremely easy to derive these monotonicity properties. We give a list of Examples.