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.
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References
Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.
Bacharach, M. (1966). Matrix rounding problems. Management Science, 12(9), 732–742.
Bertsimas, D., & Brown, D. B. (2009). Constructing uncertainty sets for robust linear optimization. Operations Research, 57(6), 1483–1495.
Cheridito, P., & Li, T. (2008). Dual characterization of properties of risk measures on Orlicz hearts. Mathematics and Financial Economics, 2(1), 29–55.
Dana, R.-A. (2005). A representation result for concave schur concave functions. Mathematical Finance, 15(4), 613–634.
Dentcheva, D., Penev, S., & Ruszczyński, A. (2010). Kusuoka representation of higher order dual risk measures. Annals of Operations Research, 181, 325–335.
Dentcheva, D., & Ruszczyński, A. (2013a). Common mathematical foundations of expected utility and dual utility theories. SIAM Journal on Optimization, 23(1), 381–405.
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.
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).
Grechuk, B., & Zabarankin, M. (2012). Schur convex functionals: Fatou property and representation. Mathematical Finance, 22(2), 411–418.
Jouini, E., Schachermayer, W., & Touzi, N. (2006). Law-invariant risk measures have the Fatou property. Advances in Mathematical Economics, 9(1), 49–71.
Kusuoka, S. (2001). On law invariant coherent risk measures. Advances in Mathematical Economics, 3, 83–95.
Leitner, J. (2005). A short note on second-order stochastic dominance preserving coherent risk measures. Mathematical Finance, 15(4), 649–651.
McNeil, A., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques, and tools. Princeton series in finance. Princeton: Princeton University Press.
Nelsen, R. B. (1999). An introduction to copulas. New York: Springer.
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.
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.
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.
Pflug, G. C., & Römisch, W. (2007). Modeling, managing and measuring risk. Singapore: World Scientific publishing.
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.
Ruszczyński, A., & Shapiro, A. (2006). Optimization of convex risk functions. Mathematics of Operations Research, 31(3), 433–452.
Shapiro, A. (2013). On Kusuoka representation of law invariant risk measures. Mathematics of Operations Research, 38, 142–152.
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.
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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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Noyan, N., Rudolf, G. Kusuoka representations of coherent risk measures in general probability spaces. Ann Oper Res 229, 591–605 (2015). https://doi.org/10.1007/s10479-014-1748-6
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DOI: https://doi.org/10.1007/s10479-014-1748-6