On the expectations of the present values of the time of ruin perturbed by diffusion

In this paper, we consider the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. A compound geometric distribution and the expectations of the present values of the time of ruin due to oscillation and a claim, respectively, are studied. Recursive formulas and explicit expressions for the moments of, and the asymptotic formulas and the Tijms-type approximations for, the compound geometric distribution and the expectations of the present values of the time of ruin are derived. In addition, explicit analytical solutions to the compound geometric distribution and to these expectations can be obtained if the claim size distribution is a combination of exponentials or a mixture of Erlangs. Finally, a lower bound and an upper bound on the compound geometric distribution are proposed, provided the associated claim size distribution is in some of the reliability-based classes.