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.