Kusuoka representations of coherent risk measures in general probability spaces
- 278 Downloads
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.
KeywordsKusuoka representation Coherent risk measures Spectral risk measures Acceptability functional Law invariance Comonotonicity
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.
- Dentcheva, D., & Ruszczyński, A. (2013b). Risk preferences on the space of quantile functions. Mathematical Programming, Ser: B (online first). doi: 10.1007/s10107-013-0724-2.
- Föllmer, H., & Schied, A. (2004). Stochastic finance. Number 27 in De Gruyter studies in mathematics. Berlin: de Gruyter, 2, rev. and extended edition.Google Scholar
- Frittelli, M., & Rosazza Gianin, E. (2005). Law invariant convex risk measures. In Kusuoka, S., Maruyama, T. (Eds.), Advances in mathematical economics (Vo. 7, pp. 33–46).Google Scholar
- McNeil, A., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques, and tools. Princeton series in finance. Princeton: Princeton University Press.Google Scholar
- Noyan, N., & Rudolf, G. (2012a). Kusuoka representations of coherent risk measures in finite probability spaces. Technical report, RUTCOR-Rutgers Center for Operations Research, RRR 33-2012. http://rutcor.rutgers.edu/pub/rrr/reports2012/33_2012.pdf.
- Noyan, N., & Rudolf, G. (2012b). Optimization with multivariate conditional value-at-risk-constraints. Technical report, Optimization online. http://www.optimization-online.org/DB_FILE/2012/04/3444.pdf.
- Noyan, N., & Rudolf, G. (2013). Optimization with multivariate conditional value-at-risk-constraints. Operations Research, 61(4), 990–1013.Google Scholar
- Pflug, G., & Wozabal, D. (2009). Ambiguity in portfolio selection. In A. H. Dempster, G. Pflug, & G. Mitra (Eds.), Quantitative fund management, Chapman & Hall/CRC financial mathematics series (pp. 377–391). Boca Raton, FL: CRC Press.Google Scholar
- Pflug, G. C. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In S. Uryasev (Ed.), Probabilistic constrained optimization: Methodology and applications. Dordrecht: Kluwer.Google Scholar
- Pichler, A., & Shapiro, A. (2012). Uniqueness of Kusuoka representations. http://www.optimization-online.org/DB_FILE/2012/10/3660.pdf.
- Rockafellar, R. T. (2007). Coherent approaches to risk in optimization under uncertainty. Tutorials in operations research, 3, 38–61.Google Scholar
- Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming: Modeling and theory. Philadelphia, USA: The society for industrial and applied mathematics and the mathematical programming society.Google Scholar