zbMATH — the first resource for mathematics

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.

62P05 Applications of statistics to actuarial sciences and financial mathematics
60G42 Martingales with discrete parameter
Full Text: DOI
[1] DOI: 10.1214/aos/1176344247 · Zbl 0389.62025
[2] Andersen P. K., Scandinavian Journal of Statistics 12 pp 97– (1985)
[3] Berger A., Assekuranz Jahrbuch 50 pp 18– (1931)
[4] Bowers N. L., Actuarial mathematics (1986) · Zbl 0634.62107
[5] Bühlmann H., Transactions of the 20th International Congress of Actuaries 5 pp 267– (1976)
[6] Gerber H. U., An introduction to mathematical risk theory (1979) · Zbl 0431.62066
[7] Gerber H. U., Lebenversicherungsmathematik (1986)
[8] Gill R. D., Censoring and stochastic integrals (1980) · Zbl 0456.62003
[9] Hattendorff K., Masius Rundschau der Versicherungen 18 pp 169– (1868)
[10] Hickman J. C., Transactions of the Society of Actuaries 16 pp 1– (1964)
[11] Hoem J. M., Blätter der Deutschen Gesellschaft für Versicherungsmathematik pp 91– (1969)
[12] Hoem J. M., Scandinavian Actuarial Journal 61 pp 38– (1978)
[13] Jacobsen M., Statistical analysis of counting processes (1982) · Zbl 0518.60065
[14] Küttner W., Mitteilungen der Öst.-Ung. Verbandes d. Privatvers.-Anst. Neue Folge 5 pp 113– (1909)
[15] Papatriandafylou A., Scandinavian Actuarial Journal 67 pp 210– (1984)
[16] Steffensen J. F., Skandinavisk Aktuarietidskrift 12 pp 1– (1929)
[17] Wolthuis H., Scandinavian Actuarial Journal 10 pp 157– (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.