The expected discounted penalty at ruin in the risk process with random income

We extend the classical risk model to the case where the premium income process, based on a Poisson process, is no longer a linear function. We examine the expected discounted value of a penalty at ruin, which is considered as a function of the initial surplus. We derive a defective renewal equation satisfied by the discounted penalty function. The solution of this renewal equation is then given. The associated compound geometric distribution is also studied.