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Mean-field approximations in insurance

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  • Philipp C. Hornung

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 approximation, 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 approximation, 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.

Suggested Citation

  • Philipp C. Hornung, 2025. "Mean-field approximations in insurance," Papers 2511.04198, arXiv.org.
    • Handle: RePEc:arx:papers:2511.04198
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    References listed on IDEAS

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    1. Francis, Laura & Steffensen, Mogens, 2024. "Individual life insurance during epidemics," Annals of Actuarial Science, Cambridge University Press, vol. 18(1), pages 152-175, March.
    2. Djehiche, Boualem & Löfdahl, Björn, 2016. "Nonlinear reserving in life insurance: Aggregation and mean-field approximation," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 1-13.
    3. Christiansen, Marcus C. & Djehiche, Boualem, 2020. "Nonlinear reserving and multiple contract modifications in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 187-195.
    4. Fahrenwaldt, Matthias A. & Weber, Stefan & Weske, Kerstin, 2018. "Pricing Of Cyber Insurance Contracts In A Network Model," ASTIN Bulletin, Cambridge University Press, vol. 48(3), pages 1175-1218, September.
    5. Marco Rehmeier & Michael Röckner, 2025. "On Nonlinear Markov Processes in the Sense of McKean," Journal of Theoretical Probability, Springer, vol. 38(3), pages 1-36, September.
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