A Representation Result for Concave Schur Concave Functions
Dana, Rose-Anne (2005), A Representation Result for Concave Schur Concave Functions, Mathematical Finance, 15, 4, p. 613-634. http://dx.doi.org/10.1111/j.1467-9965.2005.00253.x
TypeArticle accepté pour publication ou publié
Journal nameMathematical Finance
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Abstract (EN)A representation result is provided for concave Schur concave functions on L∞(Ω). In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex continuous distortion of the underlying probability. The method of proof is based on the concave Fenchel transform and on Hardy and Littlewood's inequality. Under the assumption that the probability space is nonatomic, concave, weakly upper semicontinuous, law-invariant functions are shown to coincide with weakly upper semicontinuous concave Schur concave functions. A representation result is, thus, obtained for weakly upper semicontinuous concave law-invariant functions.
Subjects / Keywordsconcave rearrangement-invariant functions; law-invariant risk measures; law-invariant concave functions; concave Schur concave functions; second-order stochastic dominance; concave order
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