## 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:

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