Hattendorff’s theorem and Thiele’s differential equation generalized. (English) Zbl 0755.62081

Summary: Hattendorff’s classical result on zero means and uncorrelatedness of the losses created in disjoint time intervals by a life insurance policy is an immediate consequence of the very definition of the concept of loss. Thus, the result is formulated and proved here in a setting with quite general payments, discount function, and time intervals, all stochastic. A general representation is given for the variances of the losses. They are easy to compute when sufficient structure is added to the model. The traditional continuous time Markov chain model is given special consideration. A stochastic Thiele’s differential equation is obtained in a fairly general counting process framework.


62P05 Applications of statistics to actuarial sciences and financial mathematics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Bremaud P., Point processes and queues (1981)
[2] Gerber H. U., An Introduction to Mathematical Risk Theory (1979) · Zbl 0431.62066
[3] DOI: 10.1007/978-3-642-71310-1
[4] Gihman I. L., The Theory of Stochastic Processes 3 (1979) · Zbl 0404.60061
[5] Hattendorff K., Masius Rundschau der Versicherungen 18 pp 169– (1868)
[6] Norberg R., Scand. Actuarial J. pp 14– (1990) · Zbl 0734.62100
[7] Norberg R., Scand. Actuarial J. pp 3– (1991) · Zbl 0759.62028
[8] Papatriandafylou A., Scand. Actuarial J. pp 210– (1984) · Zbl 0575.62087
[9] Protter P., Stochastic Integration and Differential Equations (1990) · Zbl 0694.60047
[10] Ramlau-Hansen H., Scand. Actuarial J. pp 143– (1988) · Zbl 0659.62121
[11] Wolthuis H., Scand. Actuarial J. pp 157– (1987) · Zbl 0639.62091
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