Stability properties of Haezendonck-Goovaerts premium principles

TL;DR

This paper studies the stability and continuity properties of Haezendonck-Goovaerts premium principles within Orlicz spaces, establishing conditions for dual representations and various forms of convergence.

Contribution

It provides new insights into the stability properties of Haezendonck-Goovaerts principles, including conditions for the Fatou and Lebesgue properties without restrictive assumptions.

Findings

Always satisfy the Fatou property.

Lebesgue property holds iff the Orlicz function satisfies Δ₂ condition.

Lower semicontinuity under Φ-weak convergence within the Young class.

Abstract

We investigate a variety of stability properties of Haezendonck-Goovaerts premium principles on their natural domain, namely Orlicz spaces. We show that such principles always satisfy the Fatou property. This allows to establish a tractable dual representation without imposing any condition on the reference Orlicz function. In addition, we show that Haezendonck-Goovaerts principles satisfy the stronger Lebesgue property if and only if the reference Orlicz function fulfills the so-called $\Delta_2$ condition. We also discuss (semi)continuity properties with respect to $\Phi$-weak convergence of probability measures. In particular, we show that Haezendonck-Goovaerts principles, restricted to the corresponding Young class, are always lower semicontinuous with respect to the $\Phi$-weak convergence.