The discrete stationary renewal risk model and the Gerber–Shiu discounted penalty function

This paper considers the stationary and the ordinary discrete renewal risk models. The main result is an expression of the Gerber–Shiu discounted penalty function in the stationary model in terms of the corresponding Gerber–Shiu function in the ordinary model. Subsequently, this relationship is considered in more detail in both the discount free case and under the compound binomial model. The latter case may be viewed as a discrete analog of the classical Poisson model. Simplifications of the general relationship are obtained, and a connection between the defective joint cumulative distribution functions of the surplus prior to ruin and the deficit at ruin in the stationary and the ordinary renewal risk models is established. Moreover, the defective probability function of the claim causing ruin is derived in the compound binomial case.