Original title: Method of successive approximations for solving integral equations of actuarial mathematics
Bogdan Norkin · September 2026
Insurance companies need to know the risk of running out of money, which is called ruin. The classic math model for this risk makes simple assumptions that do not always match real life. The author looked at realistic, complex scenarios, such as changing premiums, non-Poisson flows of premiums and claims, and random environments. For these cases, the author set up integral equationsEquations where the unknown function is inside an integral, used here to define ruin probabilities. to find the chance of staying solvent based on starting capital. To solve these equations, the author developed a method of successive approximations. This approach starts with a guess and improves it step by step, while also calculating upper and lower boundaries to trap the exact answer. The resulting technique lets actuaries calculate ruin probabilities to any chosen level of precision, outperforming or correcting older empirical estimates and random computer simulations.
You are packing for a hike and need to guess how much water to carry. Your first estimate is rough, but as you walk, you feel thirst and adjust, adding or removing water based on real conditions. In this paper, the method starts with an initial guess for a key insurance metric and refines it step by step using defined rules, getting closer to the exact value with each iteration until it meets any required precision.
Insurance companies need accurate ruin probability estimates to set premiums and manage risk. This method improves calculation precision, helping companies make better financial decisions and avoid unexpected losses.
The abstract does not specify the full range of models for which this method is guaranteed to work beyond those tested. It compares to Monte Carlo and known approximations but does not establish superiority in all contexts. The method relies on solving integral equations, which may not be feasible for extremely complex scenarios.
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Method of successive approximations for solving integral equations of actuarial mathematics — Bogdan Norkin
Abstract licence: CC0
Explanation generated by Synthesis from the abstract — it is not the original paper. This is a preprint, so it hasn't necessarily been peer reviewed.