A Kernel Framework for Actuarial Risk Measurement Under Heavy Tails and Dependence

Heavy-tailed claim distributions, complex dependence structures, and model uncertainty continue to challenge actuarial risk measurement and capital evaluation. Classical tools such as Value-at-Risk, Expected Shortfall, or distortion risk measures often exhibit instability under tail misspecification, while their structure provides limited flexibility for smoothing or tempering extreme losses. This paper develops a unified kernel-based framework for transforming loss distributions through measurable integral operators. Within this framework, we reinterpret the Incomplete Sadefo Transform (IST) as a specific truncated, distribution-weighted kernel operator, correcting earlier misconceptions regarding coherence and spectrality. We derive general structural properties of kernel-induced risk functionals, identifying precise conditions under which monotonicity, homogeneity, convexity, or coherence may hold, and show that spectral representations arise only in degenerate kernel configurations. We establish robustness results under Wasserstein and Kullback-Leibler perturbations, characterize heavy-tail attenuation under polynomially decaying kernels, and analyze how monotone versus non-monotone kernels affect copula-based dependence. Applications to IBNR reserving, portfolio aggregation, and capital quantification demonstrate the practical relevance of kernel transformations as pre-processing operators that moderate tail risk and dependence before standard capital assessment.