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

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  • Yuxuan Liu
  • Zhengjun Jiang
  • Yiwen Zhang

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.

Suggested Citation

  • Yuxuan Liu & Zhengjun Jiang & Yiwen Zhang, 2023. "q-scale function, Banach contraction principle, and ultimate ruin probability in a Markov-modulated jump–diffusion risk model," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2023(1), pages 38-50, January.
    • Handle: RePEc:taf:sactxx:v:2023:y:2023:i:1:p:38-50
    DOI: 10.1080/03461238.2022.2078221
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