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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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