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Nonlinear reserving in life insurance: Aggregation and mean-field approximation

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  • Djehiche, Boualem
  • Löfdahl, Björn

Abstract

We suggest a unified approach to claims reserving for life insurance policies with reserve-dependent payments driven by multi-state Markov chains. The associated prospective reserve is formulated as a recursive utility function using the framework of backward stochastic differential equations (BSDE). We show that the prospective reserve satisfies a nonlinear Thiele equation for Markovian BSDEs when the driver is a deterministic function of the reserve and the underlying Markov chain. Aggregation of prospective reserves for large and homogeneous insurance portfolios is considered through mean-field approximations. We show that the corresponding prospective reserve satisfies a BSDE of mean-field type and derive the associated nonlinear Thiele equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:69:y:2016:i:c:p:1-13
    DOI: 10.1016/j.insmatheco.2016.04.002
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    References listed on IDEAS

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    1. Cohen, Samuel N. & Elliott, Robert J., 2010. "A general theory of finite state Backward Stochastic Difference Equations," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 442-466, April.
    2. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    3. Møller,Thomas & Steffensen,Mogens, 2007. "Market-Valuation Methods in Life and Pension Insurance," Cambridge Books, Cambridge University Press, number 9780521868778, January.
    4. Marcus C. Christiansen & Michel M. Denuit & Jan Dhaene, 2014. "Reserve-Dependent Benefits and Costs in Life and Health Insurance Contracts," Tinbergen Institute Discussion Papers 14-117/IV/DSF80, Tinbergen Institute.
    5. Ragnar Norberg, 2005. "Anomalous PDEs in Markov chains: Domains of validity and numerical solutions," Finance and Stochastics, Springer, vol. 9(4), pages 519-537, October.
    6. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    7. Duffie, Darrel & Lions, Pierre-Louis, 1992. "PDE solutions of stochastic differential utility," Journal of Mathematical Economics, Elsevier, vol. 21(6), pages 577-606.
    8. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," The Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
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    Cited by:

    1. Christiansen, Marcus C. & Furrer, Christian, 2021. "Dynamics of state-wise prospective reserves in the presence of non-monotone information," Insurance: Mathematics and Economics, Elsevier, vol. 97(C), pages 81-98.
    2. Akihiro Kaneko, 2023. "Multi-stage Euler-Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains," Papers 2311.08826, arXiv.org, revised Nov 2023.
    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. Marcus C. Christiansen & Christian Furrer, 2020. "Dynamics of state-wise prospective reserves in the presence of non-monotone information," Papers 2003.02173, arXiv.org, revised Jan 2021.
    5. Marcus C. Christiansen, 2018. "A martingale concept for non-monotone information in a jump process framework," Papers 1811.00952, arXiv.org, revised Jan 2021.
    6. Kristian Buchardt & Christian Furrer & Oliver Lunding Sandqvist, 2022. "Transaction time models in multi-state life insurance," Papers 2209.06902, arXiv.org, revised Feb 2023.
    7. Marcus C. Christiansen, 2021. "Time-dynamic evaluations under non-monotone information generated by marked point processes," Finance and Stochastics, Springer, vol. 25(3), pages 563-596, July.

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