## Fourier-cosine method for ruin probabilities.(English)Zbl 1305.91163

Summary: 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 [F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput. 31, No. 2, 826–848 (2008; Zbl 1186.91214)]. 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.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 42A10 Trigonometric approximation 60E10 Characteristic functions; other transforms 62P05 Applications of statistics to actuarial sciences and financial mathematics 91G20 Derivative securities (option pricing, hedging, etc.)

Zbl 1186.91214
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### References:

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