On the Gerber–Shiu discounted penalty function in a risk model with two types of delayed-claims and random income

Abstract

This paper considers a risk model with random premium income and two types of by-claims, which is an extension to the general delayed claims process where there is only one type of by-claim and the premium income is a constant. Assume that each main claim induces one of the two by-claims, the by-claim and its associated main claim occur at the same time when the main claim amount is less than a threshold variable; otherwise, the occurrence of the by-claim will be delayed. An integral equations system for the Gerber–Shiu discounted penalty functions is presented by using auxiliary risk models. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the Gerber–Shiu penalty function is derived. By applying Rouché theorem, the Gerber–Shiu discounted penalty functions with reciprocal of the mean of premium surplus are obtained. According to Lagrange interpolation theorem, we prove that the Gerber–Shiu discounted penalty function satisfies a defective renewal equation. Finally, under the assumption of the claim sizes satisfying exponential distribution, the explicit formula of the ruin probability is derived when the discounted factor equals zero and the penalty function equals one.