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.deteriorates under heavy tails and misspecification of the copula . VaR is notoriously nonsubadditive in general, while ES is coherent but highly sensitive to model misspecification in the tail. The distortion risk measures [23, 5] and the spectral framework of Acerbi [1] and Kusuoka [15] provide elegant law-invariant extensions, but they are fundamentally based on quantile representations with fixed distribution-independent weighting functions. As a consequence, they do not easily accommodate smoothing, truncation, or tempering operations at the level of the loss distribution itself. From a practical perspective, actuaries often require operators capable of attenuating extreme tails or stabilizing dependence before capital is computed, without resorting to ad hoc caps or unrealistic moment assumptions.Kernel-based transforms as distributional operators. A natural way to modulate tail behavior and dependence at the distribution level is to apply a kernel-based integral operator to the loss distribution. Such operators unify a variety of classical transforms -including Laplace, Mellin, Stieltjes, and distortion-type transforms -under a simple measurable kernel K g (x, t). By choosing kernels that are bounded, decreasing, or horizon-dependent, actuaries can obtain transformed losses that exhibit improved robustness properties, smoother dependence, or moderated tail contributions. This idea extends the initial motivation behind the Incomplete Sadefo Transform (IST) introduced in Sadefo-Kamdem [20], in which a distribution-weighted truncated integral was proposed to retain essential tail information while reducing extreme instability. 5. Clarification of the IST methodology. We reinterpret IST as a special case of a kernel transform, correct earlier misconceptions regarding coherence and spectrality, and position IST as a flexible smoothing/tempering operator rather than a risk measure in the classical sense.Organization of the paper. Section 2 introduces the basic notation and recalls the main concepts of monetary, coherent, and spectral risk measures. Section 3 presents the general kernel-transform framework and states the structural assumptions ensuring that the operator is well defined, integrable, and stable under distributional convergence. Section 4 specializes in this framework in the Incomplete Sadefo Transform (IST). We provide a rigorous formulation of IST as a kernel-based operator, establish its existence and continuity properties, and clarify its relationship with classical integral transforms. Section 5 develops the corresponding kernelinduced and IST-based risk functionals. We derive their quantile representations, establish monotonicity and (non-)homogeneity properties, and show precisely why these operators fail, in general, to satisfy coherence or spectrality requirements. Section 6 analyzes the behavior of kernel-and IST-based risk measures under heavy-tailed loss distributions and various dependence structures. We obtain tail-attenuation results and examine how monotone and non-monotone kernels can modify copula-based dependence. It also investigates robustness under model uncertainty. We provide explicit Lipschitz-type bounds in Wasserstein distance and KL-based stability estimates derived from total variation arguments. Section 7 illustrates the actuarial relevance of the framework through an IST-driven IBNR reserving operator and kernel-based capital indicators. We establish consistency of an empirical estimator and discuss how kernel-based development factors may be incorporated into practical reserving methodologies and portfolio aggregation. Finally, Section 8 summarizes the main findings, discusses the limitations of the approach, and outlines several directions for further research, including kernel calibration, dynamic extensions, and potential connections with machine-learning architectures.