## On the ruin probabilities in a general economic environment.(English)Zbl 0997.60041

The paper analyzes the asymptotic behavior of ruin probabilities for insurance companies with large initial capital. Given that the capital $$U$$ of the company evolves in discrete time according to the rule $$U_{n}=A_{n}^{-1}(U_{n-1}-B_{n})$$, where $$A_{n}=1/(1+r_{n})$$ is the inverse of the rate of return on investment, $$B_{n}$$ is the insurance payout net of premia at date $$n$$ ($$n=1,2,...$$), the initial value of the company is $$U_{0}=M$$ and $$T_{M}=\inf\{ n\mid U_{n}<0 \}$$ is the ruin time, the author lists the conditions under which $$\lim _{M\rightarrow \infty } \log\text{Pr}\{ T_{M}<\infty \} /\log M =-w$$ for some positive $$w$$. The first group of conditions that guarantee the above asymptotic result includes independence of the return rate and the net payment processes $$A$$ and $$B$$, asymptotic growth requirements on $$A_{n}$$ plus certain uniform integrability requirements on $$B_{n}$$, and asymptotic uniform lower bound requirements on $$B_{n}$$. The second group of conditions, yielding a refinement of the general result, adds to the above named the i.i.d. property of sequences $$A$$ and $$B$$ as well as a restriction on the convolution powers of the distribution of $$\log A_{1}$$.

### MSC:

 60G40 Stopping times; optimal stopping problems; gambling theory 60F10 Large deviations
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### References:

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