q-scale function, Banach contraction principle, and ultimate ruin probability in a Markov-modulated jump–diffusion risk model

Abstract

The paper investigates ultimate ruin probability, the probability that ruin time is finite, for an insurance company whose risk reserves follow a Markov-modulated jump–diffusion risk model. We use both the Banach contraction principle and q-scale functions to prove that ultimate ruin probability is the only fixed point of a contraction mapping and show that an iterative equation can be employed to calculate ultimate ruin probability by an iterative algorithm of approximating the fixed point. Using q-scale functions and the methodology from Gajek and Rudź [(2018). Banach contraction principle and ruin probabilities in regime-switching models. Insurance: Mathematics and Economics, 80, 45–53] applied to the Markov-modulated jump–diffusion risk model, we get a more explicit Lipschitz constant in the Banach contraction principle and conveniently verify some similar results of their appendix in our case.

Keywords:

• Ruin probability
• Markov-modulated jump–diffusion risk model
• q-scale function
• Banach contraction principle

2020 Mathematics Subject Classifications:

• 91G05
• 60K37
• 60J70

Acknowledgments

The authors would like to thank the editor and referee for their very valuable comments which improved this paper.

Disclosure statement

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

Additional information

Funding

Zhengjun Jiang's work was supported in part by the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College project code 2022B1212010006 and in part by the Guangdong Higher Education Upgrading Plan (2021–2025) of ‘Rushing to the Top, Making Up Shortcomings and Strengthening Special Features’ with UIC research grant R0400001-22.