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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.

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