Entropy Based Risk Measures

TL;DR

This paper extends entropic risk measures by incorporating Rényi entropies, providing explicit relations, dual representations, and analyzing the impact of information in stochastic risk assessment.

Contribution

It introduces a generalized framework for entropic risk measures using Rényi entropies, with explicit relations and dual representations.

Findings

Explicit relations between different entropic risk measures

Characterization of dual norms and Hahn-Banach functionals

Analysis of information impact on risk measures

Abstract

Entropy is a measure of self-information which is used to quantify losses. Entropy was developed in thermodynamics, but is also used to compare probabilities based on their deviating information content. Corresponding model uncertainty is of particular interest in stochastic programming and its applications like mathematical finance, as complete information is not accessible or manageable in general. This paper extends and generalizes the Entropic Value-at-Risk by involving R\'enyi entropies. We provide explicit relations of different entropic risk measures, we elaborate their dual representations and present their relations explicitly. We consider the largest spaces which allow studying the impact of information in detail and it is demonstrated that these do not depend on the information loss. The dual norms and Hahn-Banach functionals are characterized explicitly.