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Licensed Unlicensed Requires Authentication Published by De Gruyter (A) November 13, 2014

Stochastic dominance with respect to a capacity and risk measures

Miryana Grigorova

Abstract

In our previous work, we have extended the classical notion of increasing convex stochastic dominance relation with respect to a probability to the more general case of a normalized monotone (but not necessarily additive) set function, also called a capacity. In the present paper, we pursue that work by studying the set of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to the generalized stochastic dominance relation. Under suitable assumptions on the underlying capacity space, we characterize that class of risk measures in terms of Choquet integrals with respect to a distorted capacity whose distortion function is concave. Kusuoka-type characterizations are also established. A generalization to the case of a capacity of the Tail Value at Risk is provided as an example. It is also shown that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.

The author is deeply grateful to Prof. Marie-Claire Quenez for her helpful suggestions and remarks. The comments and remarks of the two anonymous referees and the co-editor are also gratefully acknowledged.

Received: 2014-4-11
Revised: 2014-9-13
Accepted: 2014-10-29
Published Online: 2014-11-13
Published in Print: 2014-12-1

© 2014 by De Gruyter

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