Padé rational approximations are a very convenient approximation tool, due to the easiness of obtaining them, as solutions of linear systems. Not surprisingly, many matrix exponential approximations used in applied probability are particular cases of first and second order ädmissible Padé approximations” of a Laplace transform, where admissible stands for nonnegative in the case of a density, and for nonincreasing in the case of a ccdf (survival function).
Our first contribution below is the observation that for Cramér–Lundberg processes and Brownian perturbed Cramér–Lundberg processes there are three distinct rationally approximations of the Pollaczek-Khinchine transform, corresponding to approximating a) the claims transform, b) the stationary excess transform, and c) the aggregate loss transform.
A second contribution is providing three new always admissible second order approximations for the ruin probabilities of the Cramér-Lundberg process with Brownian perturbation, one of which reduces in the absence of perturbation to DeVylder’s approximation.
Our third contribution is a method for comparing the resulting approximations, based on the concept of largest weak-admissibility interval of the compounding/traffic intensity parameter ρρ.