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.

假设保险公司的风险准备金由马尔可夫调制的经典风险模型管理，其参数由有限状态不可约马尔可夫链调制。本文的主要目的是计算风险准备金第一次为负时破产时间是有限的最终破产概率。我们应用 Banach 收缩原理、q-scale 函数和 Markov 属性证明了最终破产概率是 q-scale 函数的收缩映射的不动点，并且可以通过构造迭代算法来近似计算最终破产概率固定点。与 Gajek 和 Rudź（保险：数学和经济学，80（2018），45-53）不同，我们的论文使用q在我们的例子中，在 Banach 收缩原理中获得更明确的 Lipschitz 常数的函数，因此不需要证明附录中的几个引理和定理，并且在我们的例子中证实了它们的一些假设。