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A stochastic approach to insurance cycles. (English) Zbl 0760.62095

Summary: In a series of papers T. Pentikäinen [Solvency of insurers and equalisation reserves. Helsinki (1982)] and coworkers have examined insurance cycles and their consequences for the solvency of insurers and equalization reserves. G. C. Taylor [J. D. Cummins and D. J. Derrig (eds.), Managing the insolvency risk of insurance companies, 3-76 (1991)] analyzed the underwriting cycles by means of differential equations, describing more or less deterministic behavior of the cycle. The aim of the present contribution is to give a general framework which can be applied to describe these cycles in a stochastic way. The advantage of our approach lies in the fact that we obtain analytical expressions for solvency margins and that we are able to separate the effects of pure solvency and the effect of cyclic behavior. In other words, a clear distinction is made between the classical idea of solvency and the equalization reserves due to cyclic trends. The underlying idea is from the so-called ‘path integral’, originally arising in quantum mechanics.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
60K99 Special processes
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References:

[1] Beekman, J. A., Two Stochastic Processes (1974), Wiley: Wiley New York · Zbl 0137.35601
[2] Beekman, J. A.; Fuelling, C. P., Interest and mortality randomness in some annuities, Insurance: Mathematics and Economics, 185-196 (1990) · Zbl 0711.62100
[3] (Feynman, R. P.; Hibbs, A. R., Quantum Mechanics and Path Integrals (1965), McGraw-Hill: McGraw-Hill New York) · Zbl 0176.54902
[4] Grosjean, C. C.; Goovaerts, M. J., The analytical evaluation of one-dimensional path integrals, Journal of Computational and Applied Mathematics, 311-331 (1988) · Zbl 0631.28004
[5] Pentikäinen, T., Solvency of insurers and Equalisation Reserves (1982), Insurance Publishing Company, Ltd: Insurance Publishing Company, Ltd Helsinki
[6] Taylor, G. C., (Cummins, J. D.; Derrig, A., Managing the Insolvency Risk of Insurance Companies. Managing the Insolvency Risk of Insurance Companies, An analysis of underwriting cycles and their effects on insurance solvency (1991), Kluwer: Kluwer Dordrecht), 3-76
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