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Tempering stable processes. (English) Zbl 1118.60037
Summary: A tempered stable Lévy process combines both the $$\alpha$$-stable and Gaussian trends. In a short time frame it is close to an $$\alpha$$-stable process while in a long time frame it approximates a Brownian motion. We consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization. We prove short and long time behavior of tempered stable Lévy processes and investigate their absolute continuity with respect to the underlying $$\alpha$$-stable processes. We find probabilistic representations of tempered stable processes which specifically show how such processes are obtained by cutting (tempering) jumps of stable processes. These representations exhibit $$\alpha$$-stable and Gaussian tendencies in tempered stable processes and thus give probabilistic intuition for their study. Such representations can also be used for simulation. We also develop the corresponding representations for Ornstein-Uhlenbeck-type processes.

##### MSC:
 60G52 Stable stochastic processes 60E07 Infinitely divisible distributions; stable distributions 60E10 Characteristic functions; other transforms 60G51 Processes with independent increments; Lévy processes
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##### References:
 [1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, with formulas, graphs, and mathematical tables, (1974), Dover Publications · Zbl 0515.33001 [2] Barndorff-Nielsen, O.E.; Maejima, M.; Sato, K., Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations, Bernoulli, 12, 1-33, (2006) · Zbl 1102.60013 [3] Barndorff-Nielsen, O.E.; Shephard, N., Non-Gaussian ornstein – uhlenbeck-based models and some of their uses in financial economics, J. roy. statist. soc. ser. B, 63, 1-42, (2001) · Zbl 0983.60028 [4] Barndorff-Nielsen, O.E.; Shephard, N., Normal modified stable processes, Theory probab. math. statist., 65, 1-20, (2002) [5] Bondesson, L., Generalized gamma convolutions and related classes of distributions and densities, () · Zbl 0756.60015 [6] Boyarchenko, S.; Levendorskii, S., Option pricing for truncated Lévy processes, Int. J. theor. appl. finance, 3, 549-552, (2000) · Zbl 0973.91037 [7] Brockett, P.L.; Tucker, H.G., A conditional dichotomy theorem for stochastic processes with independent increments, J. multivariate anal., 7, 13-27, (1977) · Zbl 0359.60032 [8] Carr, P.; Geman, H.; Madan, D.B.; Yor, M., The fine structure of asset returns: an empirical investigation, J. business, 75, 303-325, (2002) [9] Carr, P.; Geman, H.; Madan, D.B.; Yor, M., Stochastic volatility for Lévy processes, Math. finance, 13, 345-382, (2003) · Zbl 1092.91022 [10] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, series, and products, (2000), Academic Press · Zbl 0981.65001 [11] Jurek, Z.J.; Vervaat, W., An integral representation for selfdecomposable Banach space valued random variables, Z. wahrscheinlichkeitstheorie verw. gebiete, 62, 247-262, (1983) · Zbl 0488.60028 [12] Kallenberg, O., Foundations of modern probability, (2002), Springer-Verlag · Zbl 0996.60001 [13] Koponen, I., Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process, Phys. rev. E, 52, 1197-1199, (1995) [14] Mantegna, R.N.; Stanley, H.E., Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight, Phys. rev. lett., 73, 2946-2949, (1994) · Zbl 1020.82610 [15] Novikov, E.A., Infinitely divisible distributions in turbulence, Phys. rev. E, 50, R3303-R3305, (1994) [16] Pitman, J.; Yor, M., The two parameter poisson – dirichlet laws, Ann. probab., 25, 855-900, (1997) · Zbl 0880.60076 [17] Rosiński, J., On series representations of infinitely divisible random vectors, Ann. probab., 18, 405-430, (1990) · Zbl 0701.60004 [18] Barndorff-Nielsen, O.E.; Shephard, N., Non-Gaussian ornstein – uhlenbeck-based models and some of their uses in financial economics, J. roy. statist. soc. ser. B, 63, 167-241, (2001), 230-231 · Zbl 0983.60028 [19] Rosiński, J., Series representations of levy processes from the perspective of point processes, (), 401-415 · Zbl 0985.60048 [20] J. Rosiński, Tempered stable processes, in: O.E. Barndorff-Nielsen (Ed.), Second MaPhySto Conference on Lévy Processes: Theory and Applications. http://www.maphysto.dk/publications/MPS-misc/2002/22.pdf, 2002, pp. 215-220 [21] Samorodnitsky, G.; Taqqu, M.S., Stable non-Gaussian random processes, (1994), Chapman & Hall · Zbl 0925.60027 [22] Sato, K., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press · Zbl 0973.60001
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