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The asymptotic estimate of ruin probability under a class of risk model in the presence of heavy tails. (English) Zbl 1153.60338
Summary: In contrast with the classical Cramér-Lundberg model where the premium process is a linear function of time, we consider the ruin probability under the risk model where the aggregate premium consists of both a compound Poisson process and a linear process of time. Moreover, a constant interest force is also taken into account in our model. We restrict ourselves to the case where the claim size is heavy-tailed, i.e., the equilibrium distribution function of the claim size belongs to a wide subclass of the subexponential distribution. An asymptotic formula for the ruin probability is obtained by using the similar method of V. Kalashnikov and D. Konstantinides [Insur. Math. Econ. 27, No. 1, 145–149 (2000; Zbl 1056.60501)]. The asymptotic formula we get here is the same as the one in [S. Asmussen, Ann. Appl. Probab. 8, No. 2, 354–374 (1998; Zbl 0942.60034), C. Klüppelberg and U. Stadtmüller, Scand. Actuarial J. 1998, No. 1, 49–58 (1998; Zbl 1022.60083) and Kalashnikov and Konstantinides (loc. cit.)] which did not consider the stochastic premium.

60G05 Foundations of stochastic processes
60G50 Sums of independent random variables; random walks
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
Full Text: DOI
[1] DOI: 10.1214/aoap/1028903531 · Zbl 0942.60034 · doi:10.1214/aoap/1028903531
[2] DOI: 10.1137/S0040585X9797987 · doi:10.1137/S0040585X9797987
[3] DOI: 10.1016/S0167-6687(02)00120-8 · Zbl 1074.91027 · doi:10.1016/S0167-6687(02)00120-8
[4] DOI: 10.1016/S0167-6687(00)00045-7 · Zbl 1056.60501 · doi:10.1016/S0167-6687(00)00045-7
[5] Klüppelberg C., Scand. Actuarial J. 1 pp 49– (1998)
[6] DOI: 10.1016/S0167-6687(02)00189-0 · Zbl 1074.91029 · doi:10.1016/S0167-6687(02)00189-0
[7] Melnikov A., Risk Analysis in Finance and Insurance (2004) · Zbl 1031.91087
[8] DOI: 10.1016/0167-6687(94)00023-8 · Zbl 0838.62098 · doi:10.1016/0167-6687(94)00023-8
[9] DOI: 10.1023/B:JOTH.0000036319.21285.22 · Zbl 1065.91040 · doi:10.1023/B:JOTH.0000036319.21285.22
[10] DOI: 10.1016/j.insmatheco.2004.07.002 · Zbl 1122.91347 · doi:10.1016/j.insmatheco.2004.07.002
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