Banach contraction principle, q-scale function and ultimate ruin probability under a Markov-modulated classical risk model

Suppose that risk reserves of an insurance company are governed by a Markov-modulated classical risk model with parameters modulated by a finite-state irreducible Markov chain. The main purpose of this paper is to calculate ultimate ruin probability that ruin time, the first time when risk reserve is negative, is finite. We apply Banach contraction principle, q-scale functions and Markov property to prove that ultimate ruin probability is the fixed point of a contraction mapping in terms of q-scale functions and that ultimate ruin probability can be calculated by constructing an iterative algorithm to approximate the fixed point. Unlike Gajek and Rudź (Insurance: Mathematics and Economics, 80 (2018), 45–53), our paper uses q-scale functions to obtain more explicit Lipschitz constant in Banach contraction principle in our case so that proofs of several Lemmas and theorems in their Appendix are unnecessary and some of their assumptions are confirmed in our case.