Dual characterization of properties of risk measures on Orlicz hearts

  • Patrick Cheridito
  • Tianhui Li


We extend earlier representation results for monetary risk measures on Orlicz hearts. Then we give general conditions for such risk measures to be Gâteaux-differentiable, strictly monotone with respect to almost sure inequality, strictly convex modulo translation, strictly convex modulo comonotonicity, or monotone with respect to different stochastic orders. The theoretical results are used to analyze various specific examples of risk measures.


Risk measures Gâteaux-differentiability Strict monotonicity Strict convexity Stochastic orders Orlicz hearts 

JEL Classification

D81 G32 

Mathematics Subject Classification (2000)

91B28 91B30 46B42 


  1. 1.
    Acerbi C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Banking Fin. 26, 1505–1526 (2002)CrossRefGoogle Scholar
  2. 2.
    Acerbi C.: Coherent representation of subjective risk aversion. In: Szegö, G. (eds) Risk Measures for the 21st Century, Wiley, New York (2004)Google Scholar
  3. 3.
    Artzner P., Delbaen F., Eber J.M., Heath D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ben-Tal A., Teboulle M.: Penalty functions and duality in stochastic programming via Φ-divergence functionals. Math. Oper. Res. 12, 224–240 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cheridito P., Delbaen F., Kupper M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006)MathSciNetGoogle Scholar
  6. 6.
    Cheridito, P., Li, T.: Risk measures on Orlicz hearts. Math. Finance (2007, to appear)Google Scholar
  7. 7.
    Cherny A.: Weighted V@R and its properties. Finance Stoch. 10(3), 367–393 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cherny, A., Kupper, M.: Divergence utilities (2007, preprint)Google Scholar
  9. 9.
    Csiszar I.: On topological properties of f-divergences. Studia Sci. Math. Hungarica 2, 329–339 (1967)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Dana R.: A representation result for concave Schur concave functions. Math. Finance 15(4), 613–634 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Delbaen, F.: Coherent risk measures on general probability spaces. In: Advances in Finance and Stochastics, pp. 39–56. Springer, Berlin (2002)Google Scholar
  12. 12.
    Delbaen, F.: Risk measures for non-integrable random variables. Working Paper (2006)Google Scholar
  13. 13.
    Edgar G.A., Sucheston L.: Stopping Times and Directed Processes. Cambridge University Press, London (1992)zbMATHGoogle Scholar
  14. 14.
    Filipović D., Kupper M.: Monotone and cash-invariant convex functions and hulls. Insur Math Econ 41, 1–16 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Filipović, D., Svindland, G.: Convex risk measures beyond bounded risks, or the canonical model space for law-invariant convex risk measures is L 1. Working Paper (2008)Google Scholar
  16. 16.
    Föllmer H., Schied A.: Convex measures of risk and trading constraints. Fin. Stoch. 6(4), 429–447 (2002a)zbMATHCrossRefGoogle Scholar
  17. 17.
    Föllmer, H., Schied, A.: Robust preferences and convex measures of risk. In: Advances in Finance and Stochastics, pp. 39–56. Springer, Berlin (2002b)Google Scholar
  18. 18.
    Föllmer H., Schied A.: Stochastic Finance: An Introduction in Discrete Time. 2nd edn. Walter de Gruyter Inc., Berlin (2004)zbMATHGoogle Scholar
  19. 19.
    Frittelli M., Rosazza Gianin E.: Putting order in risk measures. J Banking Fin 26(7), 1473–1486 (2002)CrossRefGoogle Scholar
  20. 20.
    Frittelli, M., Rosazza Gianin, E.: Dynamic convex risk measures. Risk Measures for the 21st Century, chap.12, Wiley Finance, London (2004)Google Scholar
  21. 21.
    Frittelli M., Rosazza Gianin E.: Law invariant convex risk measures. Adv. Math. Econ. 7, 42–53 (2005)Google Scholar
  22. 22.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Reprint of the 1952 edition. Cambridge University Press, Cambridge (1988)Google Scholar
  23. 23.
    Jouini E., Schachermayer W., Touzi N.: Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9, 49–71 (2006)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Kusuoka S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)MathSciNetGoogle Scholar
  25. 25.
    Leitner J.: A short note on second order stochastic dominance preserving coherent risk measures. Math. Finance. 15(4), 649–651 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Müller A., Stoyan D.: Comparison Methods for Stochastic Models and Risks. In: Wiley Series in Probability and Statistics. Wiley, Chichester (2002)Google Scholar
  27. 27.
    Rockafellar R.T., Uryasev S., Zabarankin M.: Generalized deviations in risk analysis. Fin. Stoch. 10(1), 51–74 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ruszczyński A., Shapiro A.: Optimization of convex risk functions. Math. Oper. Res. 31(3), 433–452 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Schied A.: Optimal investments for risk- and ambiguity-averse preferences: a duality approach. Fin. Stoch. 11(1), 107–129 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Shaked M., Shanthikumar J.G.: Stochastic Orders. Springer, Berlin (2007)zbMATHGoogle Scholar
  31. 31.
    Zălinescu C.: Convex Analysis in General Vector Spaces. World Scientific, USA (2002)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Cambridge UniversityCambridgeUK

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