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

The discounted penalty function introduced by Gerber and Shiu [North American Actuarial Journal 2 (1998) 48] is considered in the stationary renewal risk model, where it is expressed in terms of the same discounted penalty function in the ordinary renewal risk model. This relationship unifies and generalizes known special cases. An invariance property between the stationary renewal risk model and the classical Poisson model with respect to the ruin probability is also generalized as a result.