The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion

Abstract

We consider a Sparre Andersen risk process that is perturbed by an independent diffusion process, in which claim inter-arrival times have a generalized Erlang(n) distribution (i.e. as the sum of n independent exponentials, with possibly different means). This leads to a generalization of the defective renewal equations for the expected discounted penalty function at the time of ruin given by Tsai and Willmot [10,11] and Gerber and Shiu [21,22]. The limiting behavior of the expected discounted penalty function is studied, when the dispersion coefficient goes to zero. Finally, explicit results are given for the case where n=2.

Keywords:

• Sparre Andersen risk process
• Integro-differential equation
• Defective renewal equation
• generalized Lundberg equation
• Diffusion process

Acknowledgments

The authors gratefully acknowledge the constructive comments of an anonymous referee. They helped improve the paper and clarify its presentation. This research was funded by the Society of Actuaries Ph.D. Grant Shuanming Li and the Natural Sciences and Engineering Research Council of Canada (NSERC) operating grant OGP0036860 Jose garrido.