×

zbMATH — the first resource for mathematics

Moment generating functions of compound renewal sums with discounted claims. (English) Zbl 1226.91027
Summary: The first two authors [Insur. Math. Econ. 28, No. 2, 217–231 (2001; Zbl 0988.91045); Scand. Actuarial J. 2001, No. 2, 98–110 (2001; Zbl 0979.91048)] have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen’s generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor’s, where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.
By assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.

MSC:
91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60K15 Markov renewal processes, semi-Markov processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abramowitz M., Handbook of mathematical functions (1969)
[2] Andersen E. 1957 On the collective theory of risk in case of contagion between claims Transactions of the 15th international congress of actuaries, New York, II 219 229
[3] Asmussen E., Applied probability and queues (2003) · Zbl 1029.60001
[4] DOI: 10.1016/0167-6687(92)90058-J · Zbl 0748.62058 · doi:10.1016/0167-6687(92)90058-J
[5] Bernstein D., Matrix mathematics (2005)
[6] DOI: 10.2143/AST.35.1.583170 · Zbl 1123.62013 · doi:10.2143/AST.35.1.583170
[7] DOI: 10.1016/0167-6687(88)90106-0 · Zbl 0683.62059 · doi:10.1016/0167-6687(88)90106-0
[8] Cai J., Insurance: Mathematics and Economics 7 pp 131– (2001)
[9] Cox D. R., Renewal theory (1970)
[10] DOI: 10.1016/0167-6687(87)90019-9 · Zbl 0622.62098 · doi:10.1016/0167-6687(87)90019-9
[11] Dufresne D., Scandinavian Actuarial Journal 1 pp 39– (1990)
[12] Fackrell M. W. 2003 Characterization of matrix-exponential distributions Ph.D. thesis, University of Adelaide, Australia · Zbl 1197.60013
[13] Garrido J., Encyclopedia of Actuarial Science 2 pp 875– (2004)
[14] Gerber H. U., Mitteilungen Vereinigung Schweizerische Versicherungsmathematiker 71 pp 63– (1971)
[15] DOI: 10.1111/j.0022-4367.2004.00086.x · doi:10.1111/j.0022-4367.2004.00086.x
[16] DOI: 10.1016/j.insmatheco.2006.09.006 · Zbl 1119.91054 · doi:10.1016/j.insmatheco.2006.09.006
[17] Karlin S., A first course in stochastic processes, 2. ed. (1975) · Zbl 0315.60016
[18] DOI: 10.1016/j.insmatheco.2006.07.004 · Zbl 1183.91071 · doi:10.1016/j.insmatheco.2006.07.004
[19] DOI: 10.1016/S0167-6687(00)00078-0 · Zbl 0988.91045 · doi:10.1016/S0167-6687(00)00078-0
[20] DOI: 10.1080/03461230152592755 · Zbl 0979.91048 · doi:10.1080/03461230152592755
[21] Neuts M. F., Matrix–geometric solutions in stochastic models (1981) · Zbl 0469.60002
[22] Ortega J. M., Matrix theory. A second course (1987) · doi:10.1007/978-1-4899-0471-3
[23] DOI: 10.1002/9780470317044 · doi:10.1002/9780470317044
[24] Ross S. M., Stochastic processes, 2. ed. (1996) · Zbl 0888.60002
[25] DOI: 10.1016/0167-6687(94)00023-8 · Zbl 0838.62098 · doi:10.1016/0167-6687(94)00023-8
[26] Taylor G., ASTIN Bulletin 10 pp 149– (1979) · doi:10.1017/S0515036100006474
[27] DOI: 10.2307/1426858 · Zbl 0417.60073 · doi:10.2307/1426858
[28] Wang Y. F. 2007 On the distribution of discounted compound renewal sums with PH claims M.Sc. thesis, Concordia University, Montreal
[29] Waters H., Scandinavian Actuarial Journal 2 pp 148– (1989)
[30] DOI: 10.1080/03461230410020752 · doi:10.1080/03461230410020752
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.