In the literature on risk measures, cash subadditivity was proposed to replacecash additivity, motivated by the presence of stochastic or ambiguous interest rates anddefaultable contingent claims. Cash subadditivity has been traditionally studied togetherwith quasi-convexity, in a way similar to cash additivity with convexity. In this paper, westudy cash-subadditive risk measures without quasi-convexity. One of our major results isthat a general cash-subadditive risk measure can be represented as the lower envelope of afamily of quasi-convex and cash-subadditive risk measures. Representation results of cash-subadditive risk measures with some additional properties are also examined. The notionof quasi-star-shapedness, which is a natural analogue of star-shapedness, is introduced,and we obtain a corresponding representation result via the lower envelope of normalized,quasi-convex, and cash-subadditive risk measures.

Publication:

MATHEMATICS OF OPERATIONS RESEARCH

http://dx.doi.org/10.1287/moor.2022.0312

Author:

Xia Han

School of Mathematical Sciences, Nankai University, Tianjin 300071, China

Key Laboratory of Pure Mathematics and Combinatorics,Nankai University, Tianjin 300071, China

xiahan@nankai.edu.cn

Qiuqi Wang

Maurice R. Greenberg School of Risk Science, Georgia State University, Atlanta, Georgia 30303

Corresponding autho

qwang30@gsu.edu

Ruodu Wang

Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Jianming Xia

Key Laboratory of RandomComplex Structures and Data Science, National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and SystemsScience, Chinese Academy of Sciences, Beijing 100190, China