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A Representation Result For Concave Schur Concave Functions

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  • Rose‐Anne Dana

Abstract

A representation result is provided for concave Schur concave functions on L∞(Ω). In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex continuous distortion of the underlying probability. The method of proof is based on the concave Fenchel transform and on Hardy and Littlewood's inequality. Under the assumption that the probability space is nonatomic, concave, weakly upper semicontinuous, law‐invariant functions are shown to coincide with weakly upper semicontinuous concave Schur concave functions. A representation result is, thus, obtained for weakly upper semicontinuous concave law‐invariant functions.

Suggested Citation

  • Rose‐Anne Dana, 2005. "A Representation Result For Concave Schur Concave Functions," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 613-634, October.
    • Handle: RePEc:bla:mathfi:v:15:y:2005:i:4:p:613-634
    DOI: 10.1111/j.1467-9965.2005.00253.x
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    1. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    2. Dana, R. A., 2004. "Market behavior when preferences are generated by second-order stochastic dominance," Journal of Mathematical Economics, Elsevier, vol. 40(6), pages 619-639, September.
    3. repec:dau:papers:123456789/6105 is not listed on IDEAS
    4. Dybvig, Philip H, 1988. "Distributional Analysis of Portfolio Choice," The Journal of Business, University of Chicago Press, vol. 61(3), pages 369-393, July.
    5. Alexander Schied, 2004. "On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals," Papers math/0407127, arXiv.org.
    6. repec:dau:papers:123456789/6697 is not listed on IDEAS
    7. repec:dau:papers:123456789/6771 is not listed on IDEAS
    8. repec:dau:papers:123456789/5446 is not listed on IDEAS
    9. Carlier, G. & Dana, R.-A., 2005. "Rearrangement inequalities in non-convex insurance models," Journal of Mathematical Economics, Elsevier, vol. 41(4-5), pages 483-503, August.
    10. Peleg, Bezalel, 1975. "Efficient random variables," Journal of Mathematical Economics, Elsevier, vol. 2(2), pages 243-252.
    11. repec:dau:papers:123456789/5389 is not listed on IDEAS
    12. Zilcha, Itzhak & Chew, Soo Hong, 1990. "Invariance of the efficient sets when the expected utility hypothesis is relaxed," Journal of Economic Behavior & Organization, Elsevier, vol. 13(1), pages 125-131, January.
    13. Chew, S H & Epstein, Larry G & Segal, U, 1991. "Mixture Symmetry and Quadratic Utility," Econometrica, Econometric Society, vol. 59(1), pages 139-163, January.
    14. Peleg, Bezalel & Yaari, M E, 1975. "A Price Characterization of Efficient Random Variables," Econometrica, Econometric Society, vol. 43(2), pages 283-292, March.
    15. Guillaume Carlier & Rose-Anne Dana, 2003. "Pareto efficient insurance contracts when the insurer's cost function is discontinuous," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(4), pages 871-893, June.
    16. Johannes Leitner, 2005. "A Short Note On Second‐Order Stochastic Dominance Preserving Coherent Risk Measures," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 649-651, October.
    17. Dybvig, Philip H & Ross, Stephen A, 1982. "Portfolio Efficient Sets," Econometrica, Econometric Society, vol. 50(6), pages 1525-1546, November.
    18. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
    19. Chongmin Kim, 1998. "Stochastic Dominance, Pareto Optimality, and Equilibrium Asset Pricing," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(2), pages 341-356.
    20. Atkinson, Anthony B., 1970. "On the measurement of inequality," Journal of Economic Theory, Elsevier, vol. 2(3), pages 244-263, September.
    21. repec:dau:papers:123456789/5394 is not listed on IDEAS
    22. Carlier, G. & Dana, R. A., 2003. "Core of convex distortions of a probability," Journal of Economic Theory, Elsevier, vol. 113(2), pages 199-222, December.
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