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Canonical insurance models: stochastic equations and comparison theorems

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  • Marcus C. Christiansen
  • Christian Furrer

Abstract

Thiele's differential equation explains the change in prospective reserve and plays a fundamental role in safe-side calculations and other types of actuarial model comparisons. This paper presents a `model lean' version of Thiele's equation with the novel feature that it supports any canonical insurance model, irrespective of the model's intertemporal dependence structure. The basis for this is a canonical and path-wise model construction that simultaneously handles discrete and absolutely continuous modeling regimes. Comparison theorems for differing canonical insurance models follow directly from the resulting stochastic backward equations. The elegance with which these comparison theorems handle non-equivalence of probability measures is one of their major advantages over previous results.

Suggested Citation

  • Marcus C. Christiansen & Christian Furrer, 2024. "Canonical insurance models: stochastic equations and comparison theorems," Papers 2411.12522, arXiv.org.
    • Handle: RePEc:arx:papers:2411.12522
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    References listed on IDEAS

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    1. Christian Furrer, 2022. "Scaled insurance cash flows: representation and computation via change of measure techniques," Finance and Stochastics, Springer, vol. 26(2), pages 359-382, April.
    2. Ramlau-Hansen, Henrik, 1988. "The emergence of profit in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 7(4), pages 225-236, December.
    3. 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.
    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. Ahmad, Jamaal & Bladt, Mogens & Furrer, Christian, 2023. "Aggregate Markov models in life insurance: Properties and valuation," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 50-69.
    6. 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.
    7. Andreas Niemeyer, 2015. "Safety Margins for Systematic Biometric and Financial Risk in a Semi-Markov Life Insurance Framework," Risks, MDPI, vol. 3(1), pages 1-26, January.
    8. Nießl, Alexandra & Allignol, Arthur & Beyersmann, Jan & Mueller, Carina, 2023. "Statistical inference for state occupation and transition probabilities in non-Markov multi-state models subject to both random left-truncation and right-censoring," Econometrics and Statistics, Elsevier, vol. 25(C), pages 110-124.
    9. Linnemann, Per, 1993. "On the application of Thiele's differential equation in life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 13(1), pages 63-74, September.
    10. Milbrodt, Hartmut & Stracke, Andrea, 1997. "Markov models and Thiele's integral equations for the prospective reserve," Insurance: Mathematics and Economics, Elsevier, vol. 19(3), pages 187-235, May.
    11. Kristian Buchardt & Thomas Møller & Kristian Bjerre Schmidt, 2015. "Cash flows and policyholder behaviour in the semi-Markov life insurance setup," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2015(8), pages 660-688, November.
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