Explicit Expressions for the Ruin Probabilities of Erlang Risk Processes with Pareto Individual Claim Distributions

In this paper we first consider a risk process in which claim inter-arrival times and the time until
the first claim have an Erlang (2) distribution. An explicit solution is derived for the probability of ultimate
ruin, given an initial reserve of u when the claim size follows a Pareto distribution. Follow Ramsay[8], Laplace
transforms and exponential integrals are used to derive the solution, which involves a single integral of real
valued functions along the positive real line, and the integrand is not of an oscillating kind. Then we show
that the ultimate ruin probability can be expressed as the sum of expected values of functions of two different
Gamma random variables. Finally, the results are extended to the Erlang(n) case. Numerical examples are
given to illustrate the main results.