Analysis of a threshold dividend strategy for a MAP risk model

Abstract

We consider a class of Markovian risk models in which the insurer collects premiums at rate c₁(c₂) whenever the surplus level is below (above) a constant threshold level b. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin. By interpreting that the insurer pays dividends continuously at rate c₁−c₂ whenever the surplus level is above b, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained by making use of an existing connection which links an insurer's surplus process to an embedded fluid flow process.

Keywords:

• Sparre Andersen risk model
• Phase-type distribution
• Markovian arrival process
• Laplace-Stieltjes transform
• Correlated claims
• Surplus process
• Fluid queues

Acknowledgements

The authors would like to thank the anonymous referee for the useful suggestions which helped to improve this paper. We are thankful to Dr. Soohan Ahn for a careful reading of the manuscript and for his valuable comments. This work has been assisted by the Natural Sciences and Engineering Research Council of Canada.