On the number of claims until ruin in a two-barrier renewal risk model with Erlang mixtures

In this paper, we consider the renewal risk model and we are interested in the distribution of the number νν of claims until the first time that insurer’s surplus process falls below zero (ruin) or exceeds a predefined upper barrier b>ub>u (safety level), immediately after the payment of a claim. By using exponentially tilted measures we derive an expression for the joint generating function of νν and SνSν, the surplus amount at termination time. This expression is built upon the generating functions of the overshoot and undershoot of the surplus process. Furthermore, we offer explicit results for the case where the claim amounts and the claim inter-arrival times follow mixed Erlang Distributions. We finally propose and implement an algorithm for the numerical calculation of the distributions of interest via appropriate computer algebra software.