Mathematical Programming

, Volume 148, Issue 1–2, pp 181–200 | Cite as

Risk preferences on the space of quantile functions

  • Darinka Dentcheva
  • Andrzej RuszczyńskiEmail author
Full Length Paper Series B


We propose a novel approach to quantification of risk preferences on the space of nondecreasing functions. When applied to law invariant risk preferences among random variables, it compares their quantile functions. The axioms on quantile functions impose relations among comonotonic random variables. We infer the existence of a numerical representation of the preference relation in the form of a quantile-based measure of risk. Using conjugate duality theory by pairing the Banach space of bounded functions with the space of finitely additive measures on a suitable algebra \(\varSigma \), we develop a variational representation of the quantile-based measures of risk. Furthermore, we introduce a notion of risk aversion based on quantile functions, which enables us to derive an analogue of Kusuoka representation of coherent law-invariant measures of risk.


Risk measures Quantile functions Conjugate duality  Kusuoka representation 

Mathematics Subject Classification

46N10 49N15 91B16 91B30 



We thank two anonymous referees for their insightful comments which helped improve the presentation of our results. The research of the first author was supported by the National Science Foundation Award DMS-1311978. The research of the second author was supported by the National Science Foundation Award DMS-1312016.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Department of MathematicsStevens Institute of TechnologyHobokenUSA
  2. 2.Department of Management Science and Information SystemsRutgers UniversityPiscatawayUSA

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