Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation. (English) Zbl 1144.91030

Tang, Qihe

Scand. Actuar. J. 2005, No. 1, 1-5 (2005).

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 \(-\alpha<0\).

Reviewer: A. D. Borisenko (Kyïv)

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

Keywords:

asymptotic ruin probabilities; renewal model; constant interest force; regularly varying tailed claims

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