Some results on the compound Markov binomial model

Abstract

This paper considers the compound Markov binomial risk model proposed by Cossette et al. (2003 2004). 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 2002).

Keywords:

• Compound Markov binomial
• semi-Markov
• expected discounted penalty function
• ruin probability
• deficit at ruin
• surplus just prior to ruin

This research was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU7054/04P). The research of the second author was also supported by National Natural Science Foundation of China (10571092) and the research fund for the doctoral program of higher education of China.