On a risk model with dependence between interclaim arrivals and claim sizes

We consider an extension to the classical compound Poisson risk model for which the increments of the aggregate claim amount process are independent. In Albrecher and Teugels (2006 Albrecher, H. and Teugels, J. 2006. Exponential behavior in the presence of dependence in risk theory. Journal of Applied Probability, 43(1): 257–273. Crossref], Web of Science ®] , Google Scholar]), an arbitrary dependence structure among the interclaim time and the subsequent claim size expressed through a copula is considered and they derived asymptotic results for both the finite and infinite-time ruin probabilities. In this paper, we consider a particular dependence structure among the interclaim time and the subsequent claim size and we derive the defective renewal equation satisfied by the expected discounted penalty function. Based on the compound geometric tail representation of the Laplace transform of the time to ruin, we also obtain an explicit expression for this Laplace transform for a large class of claim size distributions. The ruin probability being a special case of the Laplace transform of the time to ruin, explicit expressions are therefore obtained for this particular ruin related quantity. Finally, we measure the impact of the various dependence structures in the risk model on the ruin probability via the comparison of their Lundberg coefficients.