Markov-modulated Risk Research Articles

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.

Markov-modulated, multi-threshold dual risk model

In this paper we consider a Markov-modulated dual risk reserve process with a multi–threshold dividend strategy. We study the risk reserve process that can start at any level of the threshold. An integral equation for the conditional non–ruin probability is obtained. Further in two states, we consider the process influenced by two models. In the first model the process can be in two different states and the state can change at the arrival of a random gain with respect to the threshold barrier. In the second model the state changes at the arrival of either a random gain or an independent Poisson observer. The explicit form of the survival probabilities for both the states irrespective of the initial state being 1 or 2 and the process starting below or above the barrier are obtained when the random gain size distribution is exponential. Finally, numerical examples illustrate the main results.

Constant barrier strategies in a two-state Markov-modulated dual risk model

In this paper, we consider the dividend problem in a two-state Markov-modulated dual risk model, in which the gain arrivals, gain sizes and expenses are influenced by a Markov process. A system of integro-differential equations for the expected value of the discounted dividends until ruin is derived. In the case of exponential gain sizes, the equations are solved and the best barrier is obtained via numerical example. Finally, using numerical example, we compare the best barrier and the expected discounted dividends in the two-state Markov-modulated dual risk model with those in an associated averaged compound Poisson risk model. Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the two-state Markov-modulated dual risk model.