Moments of the present value of total dividends and related problems in the risk model with delayed claims

In this paper, a compound binomial risk model in the presence of a constant dividend barrier is studied. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim randomly and may be delayed for one time period with a certain probability. In the evaluation of the moments of the present value of dividends, the interest rates are assumed to follow a Markov chain with finite state space. A system of differential equations with certain boundary conditions satisfied by the nth moment of the present value of total dividend payments prior to ruin, given the initial environment state, is derived. We arrive at a general formula which allows us to evaluate the moments of the total discounted dividends recursively in terms of the lower order moments. Assuming the claim sizes are Kh distributed or the claim size distributions have finite support, we are able to solve for all necessary components in the general recursive formula and obtain some other results of interest. We present several numerical examples which illustrate the applicabilities of our main results and the effects of the by-claims on the moments of the present value of dividends.