Stieltjes Integration Meets Deep Learning: A Measure-Theoretic Framework for Actuarial Reserve Valuation

This paper presents a novel actuarial framework for life insurance reserve valuation,integrating Stieltjes integration with Bayesian deep learning techniques. The proposedmodel leverages measure-theoretic formalism and L´evy processes to capture both financialand demographic uncertainties, particularly in the context of systemic shocks suchas pandemics. By introducing a dual-measure compatibility condition and employingRadon-Nikodym derivatives, the framework ensures a coherent probabilistic structureacross heterogeneous data sources. Furthermore, the methodology incorporates a JSdivergence-based optimization scheme to enhance model calibration and robustness.Empirical validation is conducted using EIOPA datasets, demonstrating improved predictiveaccuracy and regulatory compliance under Solvency II. The paper also discussesthe societal implications of the model, highlighting its potential to promote fairnessand resilience in insurance systems. Author affiliation: Independent Researcher; REPRISE – Register of Expert Peer Reviewers for Italian Scientific Evaluation, Ministry of University and Research (MUR), Italy This is a revised version with expanded actuarial modeling, fairness metrics via do-calculus, Lévy-based mortality shocks, and quantum-inspired reserve valuation. Empirical validation and regulatory alignment have been significantly improved. “This is an updated version of DOI: 10.5281/zenodo.15477145”