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Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation. (English) Zbl 1144.91030
The author investigates the ruin probability of the renewal model. In this model the claims \(X_{n}\geq0\), \(n\geq1\), form a sequence of i.i.d. random variables with common distribution function \(F\), and the interarrival times, \(Y_{n}\geq0\), \(n\geq1\), form another sequence of i.i.d. random variables, which are independent of \(X_{n}, n\geq1\). Let \(N(t)=\sharp\{n\geq1:\;\tau_{n}\in[0,t]\}\), \(t\geq0\), \(\tau_{n}=\sum_{k=1}^{n}Y_{k}\), \(S(t)=\sum_{n=1}^{N(t)}X_{n}\), \(t\geq0\), and let \(C(t),\;t\geq0\) be a nonnegative and nondecreasing stochastic process, denoting the total amount of premiums accumulated up to time \(t\geq0\), let \(\delta>0\) be the constant interest force. Then the total surplus up to time \(t\), denoted by \(U(t)\), satisfies the equation \[ U(t)=xe^{\delta t}+\int_{[0,t]}e^{\delta(t-y)}C(dy)-\int_{[0,t]}e^{\delta(t-y)}S(dy),\;t\geq0. \] It is proved that under some conditions \(\psi(x)\sim{Ee^{-\delta \alpha Y_1}\over(1-Ee^{-\delta \alpha Y_1})}\bar F(x)\), where \(\psi(x)\) is the ruin probability, \(\bar F(x)=1-F(x)\) is regularly varying with index \(-\alpha<0\).

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G50 Sums of independent random variables; random walks
62E20 Asymptotic distribution theory in statistics
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References:
[1] Klüppelberg C, Scand Actuar J 1 pp 49– (1998) · Zbl 1022.60083 · doi:10.1080/03461238.1998.10413991
[2] DOI: 10.1016/0167-6687(94)00023-8 · Zbl 0838.62098 · doi:10.1016/0167-6687(94)00023-8
[3] DOI: 10.1214/aoap/1028903531 · Zbl 0942.60034 · doi:10.1214/aoap/1028903531
[4] DOI: 10.1016/S0167-6687(00)00045-7 · Zbl 1056.60501 · doi:10.1016/S0167-6687(00)00045-7
[5] DOI: 10.1016/S0167-6687(02)00189-0 · Zbl 1074.91029 · doi:10.1016/S0167-6687(02)00189-0
[6] DOI: 10.1080/03461230310017531 · Zbl 1142.62094 · doi:10.1080/03461230310017531
[7] DOI: 10.1080/15326349108807204 · Zbl 0747.60062 · doi:10.1080/15326349108807204
[8] Glukhova EV, Izv. Vyssh. Uchebn. Zaved. Fiz. 44 pp 7– (2001)
[9] Boikov AV, Veroyatnost. i Primenen. 47 pp 549– (2002) · doi:10.4213/tvp3693
[10] Petersen SS, Scand Actuar J 3 pp 147– (1989) · Zbl 0711.62097 · doi:10.1080/03461238.1989.10413865
[11] DOI: 10.2143/AST.26.1.563235 · doi:10.2143/AST.26.1.563235
[12] DOI: 10.1016/S0167-6687(01)00090-7 · Zbl 0999.91048 · doi:10.1016/S0167-6687(01)00090-7
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