Joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with Erlang(nn) claims

In this paper, we consider a delayed renewal risk model, where the first inter-arrival time can be an arbitrary positive random variable while the following inter-arrival times are assumed to be infinitely divisible, and further the claim sizes are Erlang distributed. Applying the approach of Borovkov and Dickson (2008) [7] we derive an expression for the joint density of the number of claims until ruin and the time to ruin. The special case where the claim sizes are exponentially distributed is also considered, in which the inter-arrival times are not necessarily infinitely divisible. Finally, examples of the Erlang(nn) risk model with exponential distributed claims and the Gamma risk model with Erlang(2) distributed claims are given, respectively, to illustrate the results.