On the discounted penalty function in the discrete time stationary renewal risk model

In this paper we consider the discrete time stationary renewal risk model. We express the Gerber–Shiu discounted penalty function in the stationary renewal risk model in terms of the corresponding Gerber–Shiu function in the ordinary model. In particular, we obtain a defective renewal equation for the probability generating function of ruin time. The solution of the renewal equation is then given. The explicit formulas for the discounted survival distribution of the deficit at ruin are also derived.