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.