# Kusuoka representations of coherent risk measures in general probability spaces

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## Abstract

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 a denseness result about the family of risk measures with finite Kusuoka representations.

## Keywords

Kusuoka representation Coherent risk measures Spectral risk measures Acceptability functional Law invariance Comonotonicity## Notes

### Acknowledgments

The second author has been funded by TUBITAK-2216 Research Fellowship Programme. The authors thank the Associate Editor and the anonymous referees for their valuable comments and suggestions.

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