Some results on the compound Markov binomial model

This paper considers the compound Markov binomial risk model proposed by Cossette et al. (2003 Cossette, H., Landriault, D. and Marceau, É. 2003. Ruin probabilities in the compound Markov binomial model. Scandinavian Actuarial Journal, 2003(4): 301–323. [Taylor & Francis Online] , [Google Scholar] 2004 Cossette, H., Landriault, D. and Marceau, É. 2004. Exact expressions and upper bound for ruin probabilities in the compound Markov binomial model. Insurance: Mathematics and Economics, 34: 449–466. [Crossref], [Web of Science ®] , [Google Scholar]). Two discrete-time renewal (ordinary renewal and delayed renewal) risk processes associated with the compound Markov binomial risk model are analyzed. Based on the associated ordinary renewal process, a defective renewal equation for the conditional Gerber–Shiu expected discounted penalty function is obtained. The relationship between the conditional expected discounted penalty function in the ordinary renewal case and that in the delayed renewal case is then established. From these results, the conditional ultimate probability of ruin as well as the conditional joint distribution of the surplus just prior to ruin and the deficit at ruin are studied. Finally, it is shown that a modified version of the compound Markov binomial risk model is a special case of the discrete-time semi-Markov risk model introduced by Reinhard and Snoussi (2001 Reinhard, J.M. and Snoussi, M. 2001. On the distribution of the surplus prior to ruin in a discrete semi-Markov risk model. ASTIN Bulletin, 31: 255–276. [Crossref] , [Google Scholar] 2002 Reinhard, J.M. and Snoussi, M. 2002. The severity of ruin in a discrete semi-Markov risk model. Stochastic Models, 18: 85–107. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).