Summary
We extend the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how to define such measures on the space of all random variables. We also give examples that relates the theory of coherent risk measures to game theory and to distorted probability measures. The mathematics are based on the characterisation of closed convex sets Pσ of probability measures that satisfy the property that every random variable is integrable for at least one probability measure in the set Pσ.
The author acknowledges financial support from Credit Suisse for his work and from Société Générale for earlier versions of this paper. Special thanks go to Artzner, Eber and Heath for the many stimulating discussions on risk measures and other topics. I also want to thank Maaß for pointing out extra references to related work. Discussions with Kabanov were more than helpful to improve the presentation of the paper. Part of the work was done during Summer 99, while the author was visiting Tokyo Institute of Technology. The views expressed are those of the author.
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Delbaen, F. (2002). Coherent Risk Measures on General Probability Spaces. In: Sandmann, K., Schönbucher, P.J. (eds) Advances in Finance and Stochastics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04790-3_1
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