Abstract: Ruin theory is the mainly contents of insurance mathematics, as it can supply a very useful early-warning measure for the risk of the insurance company. In this paper, we study a risk model with potentially delayed claims and random premium incomes within the framework of the compound binomial model. Using the technique of generating functions, we derive a recursive formula for the Gerber-Shiu expected discounted penalty function. Specifically, an explicit formula is obtained for the discount-free case. As applications, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.

Key words: compound binomial risk model, Gerber-Shiu expected discounted penalty function, delayed claims, random premium, recursive formula