On a compound Poisson risk model with delayed claims and random incomes

The main focus of this paper is to analyze the Gerber–Shiu penalty function of a compound Poisson risk model with delayed claims and random incomes. It is assumed that every main claim will produce a by-claim which can be delayed with a certain probability. We derive the integral equation satisfied by the Gerber–Shiu penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber–Shiu penalty function is derived. Finally, when the premium sizes have rational Laplace transforms, we also obtain the Laplace transform of the Gerber–Shiu penalty function.