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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.

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