On the discounted distribution functions for the Erlang(2) risk process

In this paper, we derive explicit expressions for the discounted joint and marginal distribution functions of the surplus immediately prior to the time of ruin and the deficit at the time of ruin, and for the discounted distribution function of the amount of the claim causing ruin, for the Erlang(2) process. We also show that these distribution functions satisfy defective renewal equations. Moreover, we find the similarity between expressions for these functions based on the Erlang(1) and Erlang(2) processes, respectively.