Fourier-cosine method for ruin probabilities

In theory, ruin probabilities in classical insurance risk models can be expressed in terms of
an infinite sum of convolutions, but its inherent complexity makes efficient computation almost impossible. In contrast, Fourier transforms of convolutions could be evaluated in a far
simpler manner. This feature aligns with the heuristic of the recently popular work by Fang
and Oosterlee, in particular, they developed a numerical method based on Fourier transform for option pricing. We here promote their philosophy to ruin theory. In this paper, we
not only introduce the Fourier-cosine method to ruin theory, which has O(n) computational
complexity, but we also enhance the error bound for our case that are not immediate from
Fang and Oosterlee (2009). We also suggest a robust method on estimation of ruin probabilities with respect to perturbation of the moments of both claim size and claim arrival
distributions. Rearrangement inequality will also be adopted to amplify the Fourier-cosine
method, resulting in an effective global estimation.