On Cramér-like asymptotics for risk processes with stochastic return on investments
The author studies a risk process following the model \(Y_t=y+P_t+\int_0^t Y_{s-} dR_s\) where \(P\) and \(R\) are Lévy processes starting off at zero. In a recent paper [Stochastic Processes Appl. 75, 135--148 (1998; Zbl 0932.60044)] the author has investigated the ruin probabilities for this process. The aim of the present paper is to develop asymptotic formulas for the ruin probability \(\psi(y)\) as the initial capital \(y\) gets large in those cases where the ruin is not certain. It is shown that \(\psi \in \text{RV}_{-\kappa_0},\) i.e., is regularly varying of order \(-\kappa_0\) in case where \(R\) dominates, here \(\kappa_0\) is the zero of \(\log E(e^{-k\widetilde{R}_1})\) with \(\widetilde{R}\) being the logarithm of the Doolean-Dade functional of \(R ,\) or in case where \(P\) dominates, here it is assumed that the Lévy measure of \(P\) is in \(\text{RV}_{-\kappa_0}\) and some further conditions are satisfied.
scientific article; zbMATH DE number 1906153