On matrix exponential approximations of ruin probabilities for the classic and Brownian perturbed Cramér–Lundberg processes

Pad... 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 the first and second order "admissible Pad... 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 Cram...r-Lundberg processes and Brownian perturbed Cram...r-Lundberg processes there are three distinct rational 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 Cram...r-Lundberg process with Brownian perturbation, one of which reduces in the absence of perturbation to De Vylder'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 ?. (ProQuest: ... denotes formulae/symbols omitted.)