Four approaches to compute the probability of ruin in the compound Poisson risk process with diffusion

This article compares four different approaches for computing the probability of ruin of the insurer compound Poisson surplus process with an additive Wiener perturbation. The first method is based on the saddlepoint approximation of asymptotic analysis. The second method is based on recursive upper and lower approximations and it is a new approach for the perturbed risk process. The third method is the fast Fourier transform. The last method is Monte Carlo importance sampling. A numerical study illustrates the high accuracy of these four methods.