Mean-field approximations in insurance

TL;DR

This paper demonstrates how mean-field models can simplify the calculation of insurance liabilities for large cohorts by replacing complex high-dimensional systems with more manageable low-dimensional equations, with convergence guarantees.

Contribution

The paper introduces a mean-field approximation framework for insurance liabilities, providing convergence results and practical examples from life and non-life insurance.

Findings

Mean-field models approximate high-dimensional insurance systems effectively.

Convergence of the approximation is established under regularity conditions.

Practical examples illustrate the applicability in life and non-life insurance.

Abstract

The calculation of the insurance liabilities of a cohort of dependent individuals in general requires the solution of a high-dimensional system of coupled linear forward integro-differential equations, which is infeasible for a larger cohort. However, by using a mean-field model, the high dimensional system of linear forward equations can be replaced by a low-dimensional system of non-linear forward integro-differential equations. We show that, subject to certain regularity conditions, the insurance liability viewed as a (conditional) expectation of a functional of an underlying jump process converges to its mean-field counterpart, as the number of individuals in the cohort goes to infinity. Examples from both life- and non-life insurance illuminate the practical importance of mean-field approximations.