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A Wiener-Hopf based approach to numerical computations in fluctuation theory for Lévy processes. (English) Zbl 1281.60045
The authors propose a numerical evaluation technique related to the fluctuation theory for Lévy processes. Here, they first rewrite transforms of interest in terms of \(K(\nu,\alpha)\) and then develop a technique to compute \(K(\nu,\alpha)\) in terms of \(\alpha\), \(\nu\) and Lévy exponent \(\phi(\, .\,)\). The authors rely on the inversion approach of P. den Iseger [Probab. Eng. Inf. Sci. 20, No. 1, 1–44 (2006; Zbl 1095.65116)] to obtain the densities and probabilities of interest. The performance of the algorithm is illustrated with various examples such as Brownian motion (with drift), a compound Poisson process, and a jump diffusion process. The paper is concluded by pointing out how the algorithm of the paper can be used in order to analyze the Lévy process.

60G51 Processes with independent increments; Lévy processes
65R10 Numerical methods for integral transforms
44A10 Laplace transform
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