Generalized Orlicz premia

2025-07-12

AI Summary

This paper addresses the mathematical modeling of generalized Orlicz premiums under non-convex loss functions. To this end, it establishes a unified theoretical framework axiomatized by cash additivity, assessability, positive homogeneity, monotonicity, and normalization. It proves that, in the non-convex setting, these axioms uniquely characterize a class of premium functionals, naturally encompassing geometric means, expectiles, and $L^p$-quantiles as special cases. A key innovation is the demonstration that cash additivity necessarily implies an $L^p$-quantile structure; moreover, under geometric convexity, the authors derive both a dual representation and a multiplicative form for generalized Orlicz premiums. By integrating functional analysis with risk measure theory, this work constitutes the first extension of Orlicz-type premiums to non-convex loss scenarios—thereby generalizing classical convex frameworks and providing novel tools for robust premium design and distribution-sensitive risk measurement.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Extend Orlicz premia to non-convex loss functions
• Analyze properties of generalized Orlicz premia
• Establish elicitable functionals for Orlicz premia

Innovation

• Generalized Orlicz premia with non-convex loss
• Dual representation for geometrically convex case
• Elicitable homogeneous monotone normalized functionals