Risk measures on Musielak-Orlicz spaces: A state-dependent perspective for insurance

This paper introduces a novel framework for risk measures defined on Musielak-Orlicz spaces, incorporating state-dependent Young functions to address the limitations of traditional uniform risk models in capturing heterogeneous tail behaviors. Extending the foundational work of Cheridito and Li [2009] on Orlicz hearts, the study establishes a robust representation theorem for convex monetary risk measures, demonstrating their expression as a maximum over penalized expectations adjusted by state-specific penalty functions. This result accommodates unbounded risks with spatially or temporally varying profiles, a critical enhancement for applications in insurance and finance where heterogeneity is prevalent. The framework subsumes coherent measures as a special case and provides a characterization of optimal probability measures, ensuring computational feasibility. Practical implications are explored through connections to insurance mathematics, including links to star-shaped risk measures, variable annuities via the Q⊙P measure, and a state-dependent generalization of the Haezendonck-Goovaerts risk measure. Additionally, an aggregation technique for portfolio risk across diverse states is proposed, accompanied by illustrative examples such as the Transformed Norm and Entropic Risk Measures. By integrating theoretical rigor with practical relevance, this study offers a versatile tool for risk assessment under complex, state-varying conditions.