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Fourier-cosine method for Gerber-Shiu functions. (English) Zbl 1314.91235

Summary: We provide a systematic study on effectively approximating the Gerber-Shiu functions, which is a hardly touched topic in the current literature, by incorporating the recently popular Fourier-cosine method. Fourier-cosine method has been a prevailing numerical method in option pricing theory since the work of F. Fang and C. W. Oosterlee [SIAM J. Sci. Comput. 31, No. 2, 826–848 (2008; Zbl 1186.91214)]. Our approximant of Gerber-Shiu functions under Lévy subordinator model has \(O(n)\) computational complexity in comparison with that of \(O(n \log n)\) via the fast Fourier transform algorithm. Also, for Gerber-Shiu functions within our proposed refined Sobolev space, we introduce an explicit error bound, which seems to be absent from the literature. In contrast with our previous work [the authors, J. Comput. Appl. Math. 281, 94–106 (2015; Zbl 1305.91163)], this error bound is more conservative without making heavy assumptions on the Fourier transform of the Gerber-Shiu function. The effectiveness of our result will be further demonstrated in the numerical studies.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
91B30 Risk theory, insurance (MSC2010)
65T40 Numerical methods for trigonometric approximation and interpolation
42A10 Trigonometric approximation
60E10 Characteristic functions; other transforms
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