On the expected penalty functions in a discrete semi-Markov risk model with randomized dividends

This paper considers the expected penalty functions for a discrete semi-Markov risk model with randomized dividends. Under the model, individual claims are governed by a Markov chain with finite state space, and the insurer pays a dividend of 1 with a probability at the end of each period if the present surplus is greater than or equal to a threshold value. Recursive formulae and the initial values for the discounted free penalty functions are derived in the two-state model. A numerical example is provided to illustrate the impact of dividend payments on ruin probabilities.