The Gerber–Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier

In this paper, we consider the classical surplus process with interest and a constant dividend barrier. Under constant interest, we derive an integro-differential equation for the Gerber–Shiu expected discounted penalty function. Following an idea of Lin, Willmot and Drekic [Lin, X.S., Willmot, G.E., Drekic, S., 2003. The classical risk model with a constant dividend barrier: Analysis of the Gerber–Shiu discounted penalty function. Insurance: Math. Econom. 33, 551–566], we obtain the solution to the integro-differential equation which is in the form of an infinite series. In some special cases with exponential claims, we are able to find closed-form expressions for the Gerber–Shiu expected discounted penalty function. Finally, we extend the integro-differential equation to the case where the surplus is invested in an investment portfolio with stochastic return on investments.