Conditional divergence risk measures

TL;DR

This paper advances the theory of conditional risk measures by developing a modular convex analysis framework, providing new representation results, and applying them to conditional entropic risk measures.

Contribution

It introduces a novel modular approach to conditional risk measures and derives a variational formula for optimized certainty equivalents in this setting.

Findings

Representation results for niveloids in conditional $L^{inity}$-space.

A variational formula for optimized certainty equivalents.

A variational representation of the conditional entropic risk measure.

Abstract

Our paper contributes to the theory of conditional risk measures and conditional certainty equivalents. We adopt a random modular approach which proved to be effective in the study of modular convex analysis and conditional risk measures. In particular, we study the conditional counterpart of optimized certainty equivalents. In the process, we provide representation results for niveloids in the conditional $L^{\infty}$-space. By employing such representation results we retrieve a conditional version of the variational formula for optimized certainty equivalents. In conclusion, we apply this formula to provide a variational representation of the conditional entropic risk measure.