Skip to main content
SpringerLink
Log in

Exact functionals and their core

  • Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

Exact cooperative games or non-additive measures, coherent lower previsions and coherent risk measures are mathematically essentially the same. They all belong to the class of exact functionals on an arbitrary set of bounded functions. We investigate the exact functionals from a functional analytic point of view, i.e. we characterize this class by a norm, present a Hahn-Banach type theorem, provide a powerful construction method and adopt the concept of the core resp. σ-core from cooperative game theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artzner, Ph., Delbaen, F., Eber, J.-M., Heath, D.: Coherent Measures of Risk. Math. Finance 3, 203–228 (1999)

    Article  MathSciNet  Google Scholar 

  2. Bonsall, F. F.: Sublinear Functionate and Ideate in Partially ordered Vector Spaces. Proc. London Math. Soc. 3, 402–418 (1954)

    Article  Google Scholar 

  3. Delbaen, F.: Convex Games and Extreme Points. J. Math. Anal. Appl. 45, 210–233 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  4. Delbaen, F.: Coherent Risk Measures on General probability Spaces. URL: http://www.math.ethz.ch/~delbaen/ftp/preprints/RiskMeasuresGen-eralSpaces.pdf (2000)

  5. Denneberg, D.: Non-Additive Measures and Integral. Kluwer, Dordrecht (1994)

    Google Scholar 

  6. Dudley, R. M.: Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove (1989)

    MATH  Google Scholar 

  7. Dunford, N., Schwartz, J. T.: Linear Operators. Interscience, New York (1958)

    MATH  Google Scholar 

  8. Gilboa, I., Schmeidler, D.: Maxmin Expected Utility with non-unique Prior. J. Math. Econ. 18, 141–153 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Huber, P. J.: Robust Statistics. Wiley, New York (1981)

    Book  MATH  Google Scholar 

  10. Kratschmer, V.: Coherence of Fuzzy Measures and their Choquet Integrals. submitted to Fuzzy Sets and Systems (2000)

  11. Maaβ, S.: Superlineare Funktionale ate Verallgemeinerung exakter kooperativer Spiele. Diplomarbeit, Universitat Bremen (2000)

    Google Scholar 

  12. Parker, J. M.: The Sigma-Core of a Cooperative Game. Manuscripta Math. 70, 247–253 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Schmeidler, D.: Cores of Exact Games I. J. Math. Anal. Appl. 40, 214–225 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  14. Schmeidler, D.: Integral Representation without Additivity. Proc. Amer. Math. Soc. 97, 255–261 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. Walley, P.: Statistical Reasoning with Imprecise probabilities. Chapman and Hall, London (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich 3, Universität Bremen, 28334, Bremen, Germany

    Sebastian Maaβ

Authors
  1. Sebastian Maaβ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastian Maaβ.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maaβ, S. Exact functionals and their core. Statistical Papers 43, 75–93 (2002). https://doi.org/10.1007/s00362-001-0087-2

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-001-0087-2

Key words

Search

Navigation