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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
Software:
EMpht
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References:
[1] Abate, J; Whitt, W, Numerical inversion of Laplace transforms of probability distributions, ORSA J Comput, 7, 36-43, (1995) · Zbl 0821.65085
[2] Asghari N, den Iseger P, Mandjes M (2012) Numerical techniques in Lévy fluctuation theory. Meth Comp Appl Probab, to appear · Zbl 1291.60096
[3] Asmussen, S; Nerman, O; Olsson, M, Fitting phase-type distributions via the EM algorithm, Scand J Stat, 23, 419-441, (1996) · Zbl 0898.62104
[4] Asmussen, S; Avram, F; Pistorius, M, Russian and American put options under exponential phase-type Lévy models, Stoch Proc Appl, 109, 79-111, (2004) · Zbl 1075.60037
[5] Asmussen, S; Madan, D; Pistorius, M, Pricing equity default swaps under an approximation to the CGMY Lévy model, J Comput Finance, 11, 79-93, (2007)
[6] Bertoin J (1998) Lévy Processes. Cambridge University Press, Cambridge · Zbl 0938.60005
[7] Carolan, C; Dykstra, R, Characterization of the least concave majorant of Brownian motion, conditional on a vertex point, with application to construction, Ann Inst Stat Math, 55, 487-497, (2003) · Zbl 1057.60077
[8] Carr, P; Geman, H; Madan, D; Yor, M, Stochastic volatility for Lévy processes, Math Finance, 13, 345-382, (2003) · Zbl 1092.91022
[9] Cooley, J; Tukey, J, An algorithm for the machine calculation of complex Fourier series, Math Comput, 19, 297-301, (1965) · Zbl 0127.09002
[10] Cont R, Tankov P (2003) Financial modelling with jump processes. Chapman & Hall/CRC Press, Boca Raton · Zbl 1052.91043
[11] Iseger, P, Numerical transform inversion using Gaussian quadrature, Probab Eng Inf Sci, 20, 1-44, (2006) · Zbl 1095.65116
[12] Iseger, P; Oldenkamp, E, Pricing guaranteed return rate products and discretely sampled Asian options, J Comput Finance, 9, 1-39, (2006)
[13] Dubner, H; Abate, J, Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform, J ACM, 15, 115-123, (1968) · Zbl 0165.51403
[14] Groeneboom, P, The concave majorant of Brownian motion, Ann Probab, 11, 1016-1027, (1983) · Zbl 0523.60079
[15] Gruntjes P, den Iseger P, Mandjes M (2012) Numerical techniques in Lévy fluctuation theory: the small-jumps case. Forthcoming · Zbl 1281.60045
[16] Harrison J (1985) Brownian motion and stochastic flow systems. Wiley, New York · Zbl 0659.60112
[17] Hazewinkel M (ed) (2001) Wiener-Hopf method. Encyclopaedia of mathematics. Springer, Berlin
[18] Kyprianou A (2006) Introductory lectures on fluctuations of Lévy processes with applications. Springer, Berlin · Zbl 1104.60001
[19] Lewis, A; Mordecki, E, Wiener-Hopf factorization for Lévy processes having positive jumps with rational transforms, J Appl Probab, 45, 118-134, (2008) · Zbl 1136.60330
[20] Nguyen-Ngoc, L; Yor, M; Aït Sahalia, Y (ed.); Hansen, LP (ed.), Exotic options and Lévy processes, (2002), Amsterdam
[21] Pecherskii, E; Rogozin, B, On the joint distribution of random variables associated with fluctuations of a process with independent increments, Theory Probab Appl, 14, 410-423, (1969)
[22] Rogers, L, Evaluating first-passage probabilities for spectrally one-sided Lévy processes, J Appl Probab, 37, 1173-1180, (2000) · Zbl 0981.60048
[23] Surya, B, Evaluating scale functions of spectrally negative Lévy processes, J Appl Probab, 45, 135-149, (2008) · Zbl 1140.60027
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