A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

Abstract

In this paper, we propose a new generalized Gerber−Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber−Shiu discounted penalty function, we construct a relation between the original Gerber−Shiu discounted penalty function and our new generalized Gerber−Shiu discounted penalty function. Next, we use Laplace transform to derive a defective renewal equation for the generalized Gerber−Shiu discounted penalty function, and give a recursive method for solving the equation. Finally, when the claim amounts obey the exponential distribution, we give some explicit expressions for the generalized Gerber−Shiu discounted penalty function. Numerical illustrations are also given to study the effect of the parameters on the generalized Gerber−Shiu discounted penalty function.

Keywords

• compound Poisson risk model
• generalized Gerber–Shiu discounted penalty function
• Laplace transform
• Dickson–Hipp operator
• recursive formula