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Kusuoka Representations of Coherent Risk Measures in Finite Probability Spaces

Kusuoka representations provide an important and useful characterization of law invariant coherent risk measures in atomless probability spaces. However, the applicability of these results is limited by the fact that such representations do not always exist in probability spaces with atoms, such as finite probability spaces. We introduce the class of functionally coherent risk measures, which allow us to use Kusuoka representations in any probability space. We show that this class contains every law invariant risk measure that can be coherently extended to a family containing all finite discrete distributions. Thus, it is possible to preserve the desirable properties of law invariant coherent risk measures on atomless spaces without sacrificing generality. We also specialize our results to risk measures on finite probability spaces, and prove that in such spaces the family of risk measures with finite Kusuoka representations is dense among all law invariant coherent risk measures over a bounded class of random variables. Acknowledgements: The first author has been supported by The Scientific and Technological Research Council of Turkey (TUBITAK) Career Award # 111M543 and the second author has been funded by TUBITAK-2216 Research Fellowship Programme.