The Gerber–Shiu discounted penalty function for classical risk model with a two-step premium rate ☆

The paper studies the expected value of a discounted penalty function for a classical risk model with a two-step premium rate. In this model, we firstly derive and solve an integro-differential equation for the Gerber-Shiu discounted penalty function, then use this result to obtain the expressions of ruin probability and the joint distribution of the surplus immediately before ruin and the deficit at ruin.