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Symmetric Gaussian mixture distributions with GGC scales. (English) Zbl 1397.60041

Summary: The aim of this study is to unify and extend hyperbolic distributions when scalars are generated from the GGC family. Such distributions play an important role for modeling asset prices. Explicit expressions of multivariate densities are presented in terms of either the Laplace transform or the density of the scalar. When scalars are members of the GGC family, then the representations are articulated with respect to the Thorin measure. Several examples are provided.

MSC:

60E07 Infinitely divisible distributions; stable distributions
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Andrews, D. F.; Mallows, C. L., Scale mixtures of normal distributions, J. Roy. Statist. Soc. Ser. B, 36, 99-102 (1974) · Zbl 0282.62017
[2] Barndorff-Nielsen, O. E., Process of normal inverse Gaussian type, Finance Stoch., 2, 41-68 (1998) · Zbl 0894.90011
[3] Barndorff-Nielsen, O. E.; Halgreen, C., Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions, Z. Wahrscheinlichkeitstheor. Verwandte Geb, 38, 41-68 (1977) · Zbl 0403.60026
[4] Bennett, B. M., On a certain multivariate non-normal distribution, Math. Proc. Cambridge Philos. Soc., 57, 434-436 (1961) · Zbl 0103.36801
[5] Blæsild, P., The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen’s bean data, Biometrika, 68, 251-263 (1981) · Zbl 0463.62048
[6] Bondesson, L., A remarkable property of generalized gamma convolutions, Probab. Theory Related Fields, 78, 321-333 (1988) · Zbl 0628.60021
[7] Bondesson, L., Generalized Gamma Convolutions and Related Classes of Distributions and Densities (1992), Springer: Springer New York · Zbl 0756.60015
[8] Cambanis, S.; Fotopoulos, S. B.; He, L., On the conditional variance for scale mixtures of normal distributions, J. Multivariate Anal., 74, 163-192 (2000) · Zbl 0962.62010
[9] Cambanis, S.; Huang, S.; Simons, G., On the theory of elliptically contoured distributions, J. Multivariate Anal., 11, 368-385 (1981) · Zbl 0469.60019
[10] Cont, R.; Tankov, P., Financial Modelling with Jump Processes (2004), Chapman & Hall/CRC: Chapman & Hall/CRC London · Zbl 1052.91043
[11] Eberlein, E.; Hammerstein, E. A., Generalized hyperbolic and inverse Gaussian distributions: Limiting cases and approximations of processes, (Seminar on Stochastic Analysis, Random Fields and Applications Vol. IV (2002), Birkhäuser: Birkhäuser New York)
[12] Eberlein, E.; Keller, U., Hyperbolic distributions in finance, Bernoulli, 1, 281-299 (1998) · Zbl 0836.62107
[13] Eberlein, E.; Keller, U.; Prause, K., New insights into smile, mispricing and value at risk: the hyperbolic model, J. Bus., 71, 371-405 (1998)
[14] Fang, K. T.; Kotz, S.; Ng, W. K., Symmetric Multivariate and Related Distributions (1990), Chapman & Hall: Chapman & Hall New York
[16] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products (2000), Academic Press: Academic Press New York · Zbl 0981.65001
[17] Graybill, F. A., Theory and Applications in Linear Models (1976), Duxbury · Zbl 0121.35605
[18] James, L. F.; Roynette, B.; Yor, M., Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples, Probab. Surv., 5, 346-415 (2008) · Zbl 1189.60035
[19] Kano, Y., Consistency property of elliptical probability density functions, J. Multivariate Anal., 51, 139-147 (1994) · Zbl 0806.62039
[20] Kelker, D., Distribution theory of spherical distributions and a location scale parameter, Sankhyā Ser. A, 32, 419-430 (1970) · Zbl 0223.60008
[21] Kelker, D., Infinite divisibility and variance mixtures of the normal distribution, Ann. Math. Statist., 42, 802-808 (1971) · Zbl 0234.60009
[22] Kingman, J. F.C., On random sequences with spherical symmetry, Biometrika, 59, 492-498 (1972) · Zbl 0238.60025
[23] Kou, S. G., A jump diffusion model for option pricing with three properties: leptokurtic feature, volatility smile, and analytical tractability, Manage. Sci., 48, 1087-1101 (2002)
[24] Lange, K. L.; Sinsheimer, J. S., Normal/independent distributions and their applications in robust regression, J. Comput. Graph. Statist., 17, 81-92 (1993)
[25] Madan, D. B.; Seneta, E., The variance gamma (V.G.) model for share market returns, J. Bus., 63, 511-524 (1990)
[26] Owen, J.; Rabinovitch, R., On the class of elliptical distributions and their applications to the theory of portfolio choice, J. Finance, 38, 745-752 (1983)
[27] Samorodnitsky, G.; Taqqu, M., Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0925.60027
[28] Sato, K., Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press: Cambridge University Press New York · Zbl 0973.60001
[29] Schoenberg, I. J., Metric spaces and completely monotone functions, Ann. of Math., 24, 811-841 (1938) · Zbl 0019.41503
[30] Yano, K.; Yano, Y.; Yor, M., On the laws of the first hitting of points for one-dimensional symmetric stable Lévy processes, (Séminaire De Probabilités XLII, Vol. 1979 (2008), Springer)
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