How long does the surplus stay close to its historical high?

Abstract

In this paper we find the Laplace transforms of the weighted occupation times for a spectrally negative Lévy surplus process to spend below its running maximum up to the first exit times. The results are expressed in terms of generalized scale functions. For step weight functions, the Laplace transforms can be further expressed in terms of scale functions.

Keywords:

• Spectrally negative Lévy risk process
• occupation time
• scale function

2010 Mathematics Subject Classifications:

• 60J45
• 60J75

Acknowledgments

We thank anonymous referees for very helpful suggestions. Yun Hua and Bo Li thank Concordia University where the first draft of this paper was completed during their visits in early 2017. Xiaowen Zhou thanks National Center for Theoretical Sciences (NCTS) where this paper was revised during his visit.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [RGPIN-2016-06704].