Generalized Orlicz premia

TL;DR

This paper introduces a broad generalization of Orlicz premia using non-convex loss functions, unifying various examples like geometric mean and expectiles, and explores their properties, dual representations, and elicitable nature.

Contribution

It extends the classical theory of Orlicz premia to non-convex settings, providing new insights into their structure, duality, and elicitation properties.

Findings

Generalized Orlicz premia include geometric mean and expectiles.

Cash-additivity characterizes $L^p$-quantiles, extending classical results.

In the geometrically convex case, dual representations are characterized and compared.

Abstract

We introduce a generalized version of Orlicz premia, based on possibly non-convex loss functions. We show that this generalized definition covers a variety of relevant examples, such as the geometric mean and the expectiles, while at the same time retaining a number of relevant properties. We establish that cash-additivity leads to $L^p$-quantiles, extending a classical result on 'collapse to the mean' for convex Orlicz premia. We then focus on the geometrically convex case, discussing the dual representation of generalized Orlicz premia and comparing it with a multiplicative form of the standard dual representation for the convex case. Finally, we show that generalized Orlicz premia arise naturally as the only elicitable, positively homogeneous, monotone and normalized functionals.