Refinements of Kusuoka representations on L∞. (2nd November 2022)

• Record Type: Journal Article
• Authors: Uğurlu, Kerem
• Is Part Of: Optimization. Volume 71:Number 11(2022)
• Page Start: 3351
• Page End: 3362
• Publication Date: 2022-11-02
• Subjects: Coherent risk measures -- law invariance -- average value-at-risk -- comonotonic risk measures -- robust performance measures; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2022.2038152
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : We study Kusuoka representations of law-invariant coherent risk measures on the space of bounded random variables, which says that any law-invariant coherent risk measure is the supremum of integrals of Average-Value-at-Risk measures. We refine this representation by showing that the supremum in Kusuoka representation is attained for some probability measure in the unit interval. Namely, we prove that any law-invariant coherent risk measure on the space of bounded random variables can be written as an integral of the Average-Value-at-Risk measures on the unit interval with respect to some probability measure. This representation gives a numerically constructive way to bound any law-invariant coherent risk measure on the space of essentially bounded random variables from above and below. The results are illustrated on specific law-invariant coherent risk measures along with numerical simulations.