A note on robust representations of law-invariant quasiconvex functions

  • Samuel Drapeau
  • Michael Kupper
  • Ranja Reda
Part of the Advances in Mathematical Economics book series (MATHECON, volume 15)


We give robust representations of law-invariant monotone quasiconvex functions. The results are based on Jouini et al. (Adv Math Econ 9:49–71, 2006) and Svindland (Math Financ Econ, 2010), showing that law-invariant quasiconvex functions have the Fatou property.

Key words

Fatou property law-invariance risk measure robust representation 


  1. 1.
    Artzner, Ph., Delbaen, F., Eber, J.M., Heath, D.: Coherent risk measures. Math. Finance 9(3), 203–228 (1999)CrossRefGoogle Scholar
  2. 2.
    Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk measures: rationality and diversification. Math. Finance (2010)Google Scholar
  3. 3.
    Delbaen, F.: Coherent measure of risk on general probability spaces. In: Sandmann, K., Schonbucher, P.J. (eds.) Advances in Finance and Stochastics, Essays in Honor of Dieter Sondermann, pp. 1–37. Springer, Berlin (2002)Google Scholar
  4. 4.
    Drapeau, S., Kupper, M.: Risk preferences and their robust representation. Preprint (SSRN) (2010)Google Scholar
  5. 5.
    El Karoui, N., Ravanelli, C.: Cash sub-additive risk measures and interest rate ambiguity. Math. Finance 19(4), 561–590 (2008)CrossRefGoogle Scholar
  6. 6.
    Filipović, D., Svindland, G.: The canonical model space for law-invariant convex risk measures is L 1. Math. Finance (2008)Google Scholar
  7. 7.
    Föllmer, H., Schied, A.: Convex measure of risk and trading constraints. Finance Stoch. 6, 429–447 (2002)CrossRefGoogle Scholar
  8. 8.
    Föllmer, H., Schied, A.: Stochastic finance, an introduction in discrete time. de Gruyter Studies in Mathematics 27 (2002)Google Scholar
  9. 9.
    Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Finance 26(7), 1473–1486 (2002)Google Scholar
  10. 10.
    Frittelli, M., Rosazza Gianin, E.: Law-invariant convex risk measures. Adv. Math. Econ. 7, 33–46 (2005)Google Scholar
  11. 11.
    Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9, 49–71 (2006)CrossRefGoogle Scholar
  12. 12.
    Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math. Financ. Econ. 2(3), 189–210 (2009)CrossRefGoogle Scholar
  13. 13.
    Kusuoka, S.: On law-invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)Google Scholar
  14. 14.
    Svindland, G.: Continuity properties of law-invariant (quasi-)convex risk functions. Math. Financ. Econ. 1, 39–43 (2010)CrossRefGoogle Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Humboldt University BerlinBerlinGermany
  2. 2.Vienna University of TechnologyViennaAustria

Personalised recommendations