Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems

TL;DR

This paper develops statistical methods for estimating complex risk functionals, including their asymptotic properties, to improve risk measurement and optimization in finance and insurance.

Contribution

It introduces central limit theorems for composite risk functionals and analyzes the asymptotic behavior of related optimization problems.

Findings

Established CLT for composite risk functionals.

Analyzed asymptotic behavior of risk-based optimization.

Applicable to coherent and more general risk measures.

Abstract

We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit formulae for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest.