2012/10/26 by Alois Pichler, Alexander Shapiro, Pichler, Alois +1
Decision Sciences · #FOS: Economics and business #FOS: Mathematics #Probabilistic and Robust Engineering Design #Probability (math.PR) #Probability and Risk Models #Risk Management (q-fin.RM) #Risk and Portfolio Optimization #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1210.7257
openalex publication_date 2012/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka representation is derived from this initial result. Further, stochastic order relations are employed to identify the minimal Kusuoka representation. It is shown that measures in the minimal representation are extremal with respect to the order relations. The tools are finally employed to provide the minimal representation for important practical examples. Although the Kusuoka representation is usually given only for nonatomic probability spaces, this presentation closes the gap to spaces with atoms.