Two-sided Bounds for Renewal Equations and Ruin Quantities

In this paper, the objective is to provide sequences of improved non-increasing (non-decreasing) upper (lower) bounds for the solution of (defective) renewal equations in terms of the right-tail probability of a compound geometric distribution. Exponential (Lundberg type) and non-exponential type bounds are also derived. Also, under several reliability classifications, some new as well as improvements of well-known bounds are given. The results are then applied to obtain refinements of the bounds for several ruin related quantities, (such as the deficit at ruin, the joint distribution of the surplus prior to and at ruin, the mean deficit at ruin and the stop-loss premium, and the compound geometric densities). Bounds for the renewal function are also given.