Numerical evaluation of ruin probabilities for a finite period

In this paper the authors remind of the known formulas for the double Laplace-Stieltjes transforms of the ruin probabilities ψ(u, t), where u is the initial risk reserve and t stands for the operational time, in the case of independent interoccurence times and claim amounts such that the interoccurrence times are identically distributed Κ(t), t ≥ o, Κ(o) = o, and the claim amounts are identically distributed P(y), — ∞ < y < ∞. For some cases, where I — P(y) and I — Κ(t) are exponential polynomials, numerical inversions of the said Laplace-Stieltjes transforms are made for a selection of u- and t-values in combination with safety loadings of various sizes and signs. Moreover, some values are given when I — P(y) or I — Κ(t) are of Pareto type and comparisons are made with the results when the Pareto distributions are approximated by suitable exponential polynomials.