Expected discounted penalty function of Markov-Modulated, Constant barrier and Stochastic income model

This article considers a Markov -modulated risk process with stochastic premium income and a constant dividend barrier. We derive the integro-differential equations satisfied by the expected discounted penalty function regulated by an external environment. Applications of the integral equations are given to be the discounted expectation of the deficit at ruin. Explicit solution of the expected deficit at ruin for the model is obtained for exponential distribution. Finally in two state model, numerical example illustrates the effect of the different parameters.