On Conditional Chisini Means and Risk Measures

TL;DR

This paper explores the existence of conditional nonlinear means, called Conditional Chisini means, and applies these findings to characterize scalarizations of conditional risk measures, advancing the mathematical understanding of risk assessment tools.

Contribution

It introduces sufficient conditions for the existence and uniqueness of Conditional Chisini means, extending the theory of nonlinear means and their application to risk measures.

Findings

Established existence and uniqueness conditions for Conditional Chisini means.

Characterized scalarizations of conditional Risk Measures using these means.

Provided a mathematical framework linking nonlinear means to risk measure dual representations.

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.