Drawdown analysis for the renewal insurance risk process

Abstract

In this paper, we study some drawdown-related quantities in the context of the renewal insurance risk process with general interarrival times and phase-type distributed jump sizes. We make use of some recent results on the two-sided exit problem for the spectrally negative Markov additive process and a fluid flow analogy between certain queues and risk processes to solve for the two-sided exit problem of the renewal insurance risk process. The two-sided exit quantities are later shown to be central to the analysis of drawdown quantities including the drawdown time, the drawdown size, the running maximum (minimum) at the drawdown time, the last running maximum time prior to drawdown, the number of jumps before drawdown and the number of excursions from running maximum before drawdown. Finally, we consider another application of our methodology for the study of the expected discounted dividend payments until ruin.

Keywords:

• drawdown
• two-sided exit problem
• fluid flow technique
• renewal insurance risk process
• constant dividend barrier strategy
• ruin probability

Acknowledgements

The authors would like to thank two anonymous referees for their helpful comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by individual grants by the Natural Sciences and Engineering Research Council of Canada to David Landriault [grant number 341316] and Bin Li [grant number 05828]. Support from a start-up grant from the University of Waterloo is acknowledged by Bin Li, as is support from the Canada Research Chair program by David Landriault. Shu Li acknowledges the support from the James C. Hickman Scholar program of the Society of Actuaries, and from the Waterloo Research Institute in Insurance, Securities and Quantitative Finance.