Bregman superquantiles. Estimation methods and applications

TL;DR

This paper introduces the Bregman superquantile, extending risk measures using Bregman divergences, and analyzes its properties and estimation methods for applications in risk management.

Contribution

It defines the Bregman superquantile, studies its coherence properties, and investigates the asymptotic behavior of Monte Carlo estimators for this new risk measure.

Findings

Bregman superquantile generalizes existing risk measures.

The paper establishes coherence properties of the Bregman superquantile.

Asymptotic properties of Monte Carlo estimators are analyzed.

Abstract

In this work, we extend some quantities introduced in "Optimization of conditional value-at-risk" of R.T Rockafellar and S. Uryasev to the case where the proximity between real numbers is measured by using a Bregman divergence. This leads to the definition of the Bregman superquantile. Axioms of a coherent measure of risk discussed in "Coherent approches to risk in optimization under uncertainty" of R.T Rockafellar are studied in the case of Bregman superquantile. Furthermore, we deal with asymptotic properties of a Monte Carlo estimator of the Bregman superquantile.