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Differential equations for moments of present values in life insurance. (English) Zbl 0836.62088

Summary: Ordinary differential equations are obtained for first and higher order conditional moments of present values of payments in respect of a life insurance policy described as a time-continuous Markov chain. Those for the first moments are the well-known Thiele’s differential equations for the reserve. It is shown how the differential equations can be used to construct untraditional insurance products. Numerical computations of moments are performed for some forms of insurance common in practice. Applications to problems in pure probability theory are demonstrated by examples.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J27 Continuous-time Markov processes on discrete state spaces
65C99 Probabilistic methods, stochastic differential equations
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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References:

[1] Andersen, P.K.; Borgan, Ø.; Gill, R.D.; Keiding, N., Statistical models based on counting processes, (1993), Springer-Verlag New York, Berlin, Heidelberg · Zbl 0769.62061
[2] Hoem, J.M., Markov chain models in life insurance, Blätter der deutschen gesellschaft für versicherungsmathematik, 9, 91-107, (1969) · Zbl 0191.51103
[3] Hoem, J.M.; Aalen, O.O., Actuarial values of payment streams, Scandinavian actuarial journal, 1978, 38-47, (1978) · Zbl 0377.62067
[4] Norberg, R., Reserves in life and pension insurance, Scandinavian actuarial journal, 1991, 3-24, (1991) · Zbl 0759.62028
[5] Norberg, R., Hattendorff’s theorem and Thiele’s differential equation generalized, Scandinavian actuarial journal, 1992, 2-14, (1992), (Addendum to appear.) · Zbl 0755.62081
[6] Norberg, R., A time-continuous Markov chain interest model with applications to insurance, Journal of applied stochastic models and data analysis, (1995), To appear in · Zbl 1067.91509
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