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

This research develops a novel measure-theoretic framework integrating Stieltjes integrationwith Bayesian deep learning for life insurance reserve valuation. Building on deFinetti’s subjective probability and Rudin’s functional analysis, we reformulate actuarialreserves via Radon-Nikodym derivatives to jointly model financial and demographic risks.The framework combines measure-adapted neural operators, Lévy-process foundations,and quantum-inspired computational accelerations. Rigorous benchmarking on EIOPAmortality data shows that our Bayesian neural network reduces mean absolute error by 63%compared to traditional methods, with a stochastic MLP variant achieving 75% faster convergence.Theoretical contributions include a generalized Stieltjes operator for Lévy-typeprocesses and information-geometric regularization of neural networks. Empirical resultsdemonstrate 77% KL divergence reduction and alignment with Lévy 𝛼 = 1.7 tail behavior,confirming robustness to extremal events. A projected 𝑂(𝑁2) complexity reduction viaGrover’s algorithm enables real-time reserving for large portfolios, offering transformativepotential for Solvency II compliance. textbfEthical Statement: This study adopts certifiedethical protocols for fairness, transparency, and regulatory compliance, with AI-assistedelements rigorously validated by domain experts. Author affiliation: Independent Researcher; REPRISE – Register of Expert Peer Reviewers for Italian Scientific Evaluation, Ministry of University and Research (MUR), Italy This is a preprint deposited on Zenodo. The final version may differ.