On the ruin probabilities in a general economic environment (Q1613645)

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 \(\log A_{1}\).