Robust Preferences and Convex Measures of Risk

  • Hans Föllmer
  • Alexander Schied


We prove robust representation theorems for monetary measures of risk in a situation of uncertainty, where no probability measure is given a priori. They are closely related to a robust extension of the Savage representation of preferences on a space of financial positions which is due to Gilboa and Schmeidler. We discuss the problem of computing the monetary measure of risk induced by the subjective loss functional which appears in the robust Savage representation.


Probability Measure Penalty Function Risk Measure Financial Position Translation Invariance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. Coherent measures of risk. Math. Finance 9, no. 3, 203–228 (1999).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Choquet, G. Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1955).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Delbaen, F. Coherent measures of risk on general probability spaces. Preprint ETH Zürich (2000).Google Scholar
  4. 4.
    Delbaen, F. Coherent risk measures. Lecture notes, Pisa (2001).Google Scholar
  5. 5.
    Denneberg, D. Non-additive measure and integral. Kluwer Academic Publishers (1994).Google Scholar
  6. 6.
    Dunford, N., Schwartz, J. Linear Operators. Part I: General Theory. New York: Interscience Publishers, 1958.Google Scholar
  7. 7.
    Föllmer, H. Schied, A. Convex measures of risk and trading constraints. Finance Stoch. (to appear).Google Scholar
  8. 8.
    Föllmer, H., Schied, A. Martingale methods in mathematical finance: an introduction in discrete time. Berlin, de Gruyter Studies in Mathematics (to appear).Google Scholar
  9. 9.
    Heath, D. Back to the future. Plenary lecture at the First World Congress of the Bachelier Society, Paris 2000.Google Scholar
  10. 10.
    Huber, P., Strassen, V. Minimax tests and the Neyman-Pearson lemma for capacities. Ann. Statistics 1, no. 2, 251–263 (1973).MATHCrossRefGoogle Scholar
  11. 11.
    Gilboa, I. Expected utility with purely subjective non-additive probabilities. J. Math. Econ. 16, 65–88 (1987).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gilboa, I., Schmeidler, D. Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989).MathSciNetMATHGoogle Scholar
  13. 13.
    Huber, P. Robust statistics. New York: Wiley (1981).MATHCrossRefGoogle Scholar
  14. 14.
    Jarrow, R. Put Option Premiums and Coherent Risk Measures. Mathematical Finance (to appear)Google Scholar
  15. 15.
    Karni, E., Schmeidler, D. Utility theory with uncertainty. In: W. Hildebrandt and H. Sonnenschein (eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier Science Publishers (1991).Google Scholar
  16. 16.
    Savage, L. J. The foundations of statistics. New York: John Wiley and Sons (1954).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Hans Föllmer
    • 1
  • Alexander Schied
    • 2
  1. 1.Institut of MathematicsHumboldt-UniversityBerlinGermany
  2. 2.Institute of MathematicsTechnische UniversitätBerlinGermany

Personalised recommendations