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Hattendorff’s theorem: A Markov chain and counting process approach. (English) Zbl 0659.62121
The paper provides a generalization of Hattendorff’s theorem [K. Hattendorff, Das Risiko bei der Lebensversicherung. Masius Rundschau der Versicherungen 18, 169-183 (1868)] to the situation where a life insurance policy is modelled as a time-inhomogeneous Markov chain with a finite state space. It is shown that the present values of the gains obtained in the different states are zero mean martingales and that gains realized in different states are uncorrelated. Moreover, variance formulas for the present values of the gains are derived, and the results are illustrated by examples relating to term and disability insurance.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G42 Martingales with discrete parameter
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