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Dual characterization of properties of risk measures on Orlicz hearts

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Abstract

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.

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References

  1. Acerbi C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Banking Fin. 26, 1505–1526 (2002)

    Article  Google Scholar 

  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. Artzner P., Delbaen F., Eber J.M., Heath D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ben-Tal A., Teboulle M.: Penalty functions and duality in stochastic programming via Φ-divergence functionals. Math. Oper. Res. 12, 224–240 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheridito P., Delbaen F., Kupper M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006)

    MathSciNet  Google Scholar 

  6. Cheridito, P., Li, T.: Risk measures on Orlicz hearts. Math. Finance (2007, to appear)

  7. Cherny A.: Weighted V@R and its properties. Finance Stoch. 10(3), 367–393 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cherny, A., Kupper, M.: Divergence utilities (2007, preprint)

  9. Csiszar I.: On topological properties of f-divergences. Studia Sci. Math. Hungarica 2, 329–339 (1967)

    MATH  MathSciNet  Google Scholar 

  10. Dana R.: A representation result for concave Schur concave functions. Math. Finance 15(4), 613–634 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Delbaen, F.: Coherent risk measures on general probability spaces. In: Advances in Finance and Stochastics, pp. 39–56. Springer, Berlin (2002)

  12. Delbaen, F.: Risk measures for non-integrable random variables. Working Paper (2006)

  13. Edgar G.A., Sucheston L.: Stopping Times and Directed Processes. Cambridge University Press, London (1992)

    MATH  Google Scholar 

  14. Filipović D., Kupper M.: Monotone and cash-invariant convex functions and hulls. Insur Math Econ 41, 1–16 (2007)

    Article  MATH  Google Scholar 

  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)

  16. Föllmer H., Schied A.: Convex measures of risk and trading constraints. Fin. Stoch. 6(4), 429–447 (2002a)

    Article  MATH  Google Scholar 

  17. Föllmer, H., Schied, A.: Robust preferences and convex measures of risk. In: Advances in Finance and Stochastics, pp. 39–56. Springer, Berlin (2002b)

  18. Föllmer H., Schied A.: Stochastic Finance: An Introduction in Discrete Time. 2nd edn. Walter de Gruyter Inc., Berlin (2004)

    MATH  Google Scholar 

  19. Frittelli M., Rosazza Gianin E.: Putting order in risk measures. J Banking Fin 26(7), 1473–1486 (2002)

    Article  Google Scholar 

  20. Frittelli, M., Rosazza Gianin, E.: Dynamic convex risk measures. Risk Measures for the 21st Century, chap.12, Wiley Finance, London (2004)

  21. Frittelli M., Rosazza Gianin E.: Law invariant convex risk measures. Adv. Math. Econ. 7, 42–53 (2005)

    Google Scholar 

  22. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Reprint of the 1952 edition. Cambridge University Press, Cambridge (1988)

  23. Jouini E., Schachermayer W., Touzi N.: Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9, 49–71 (2006)

    Article  MathSciNet  Google Scholar 

  24. Kusuoka S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)

    MathSciNet  Google Scholar 

  25. Leitner J.: A short note on second order stochastic dominance preserving coherent risk measures. Math. Finance. 15(4), 649–651 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Müller A., Stoyan D.: Comparison Methods for Stochastic Models and Risks. In: Wiley Series in Probability and Statistics. Wiley, Chichester (2002)

  27. Rockafellar R.T., Uryasev S., Zabarankin M.: Generalized deviations in risk analysis. Fin. Stoch. 10(1), 51–74 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ruszczyński A., Shapiro A.: Optimization of convex risk functions. Math. Oper. Res. 31(3), 433–452 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Schied A.: Optimal investments for risk- and ambiguity-averse preferences: a duality approach. Fin. Stoch. 11(1), 107–129 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Shaked M., Shanthikumar J.G.: Stochastic Orders. Springer, Berlin (2007)

    MATH  Google Scholar 

  31. Zălinescu C.: Convex Analysis in General Vector Spaces. World Scientific, USA (2002)

    MATH  Google Scholar 

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Correspondence to Patrick Cheridito.

Additional information

We thank Andreas Hamel and Michael Kupper for fruitful discussions and helpful comments. P. Cheridito has been supported by NSF Grant DMS-0642361, a Rheinstein Award and a Peek Fellowship. T. Li has been supported by a Marshall Scholarship and a Merage Fellowship.

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Cheridito, P., Li, T. Dual characterization of properties of risk measures on Orlicz hearts. Math Finan Econ 2, 29–55 (2008). https://doi.org/10.1007/s11579-008-0013-7

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  • DOI: https://doi.org/10.1007/s11579-008-0013-7

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