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On Conditional Chisini Means and Risk Measures

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  • Alessandro Doldi
  • Marco Maggis

Abstract

Given a real valued functional T on the space of bounded random variables, we investigate the problem of the existence of a conditional version of nonlinear means. We follow a seminal idea by Chisini (1929), defining a mean as the solution of a functional equation induced by T. We provide sufficient conditions which guarantee the existence of a (unique) solution of a system of infinitely many functional equations, which will provide the so called Conditional Chisini mean. We apply our findings in characterizing the scalarization of conditional Risk Measures, an essential tool originally adopted by Detlefsen and Scandolo (2005) to deduce the robust dual representation.

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  • Alessandro Doldi & Marco Maggis, 2022. "On Conditional Chisini Means and Risk Measures," Papers 2209.10871, arXiv.org.
    • Handle: RePEc:arx:papers:2209.10871
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    References listed on IDEAS

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    1. Gerard Debreu, 1959. "Topological Methods in Cardinal Utility Theory," Cowles Foundation Discussion Papers 76, Cowles Foundation for Research in Economics, Yale University.
    2. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    3. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and Dynamic Convex Risk Measures," SFB 649 Discussion Papers SFB649DP2005-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Peter P. Wakker & Horst Zank, 1999. "State Dependent Expected Utility for Savage's State Space," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 8-34, February.
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