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Approximating Multivariate Tempered Stable Processes

Part of: Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Boris Baeumer  and
Mihály Kovács
Boris Baeumer*
Affiliation:
University of Otago
Mihály Kovács*
Affiliation:
University of Otago
*
∗ Postal address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand.
∗ Postal address: Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin, New Zealand.
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Abstract

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We give a simple method to approximate multidimensional exponentially tempered stable processes and show that the approximating process converges in the Skorokhod topology to the tempered process. The approximation is based on the generation of a random angle and a random variable with a lower-dimensional Lévy measure. We then show that if an arbitrarily small normal random variable is added, the marginal densities converge uniformly at an almost linear rate, providing a critical tool to assess the performance of existing particle tracking codes.

Keywords

Tempered stable processsimulationLévy processoperator stable processspectral measure

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G52: Stable processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 167 - 183
Copyright
© Applied Probability Trust 

References

Baeumer, B. and Meerschaert, M. M. (2010). Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448.Google Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–332.Google Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13, 345–382.Google Scholar
Cartea, A. and del Castillo-Negrete, D. (2007). Fluid limit of the continuous-time random walk with general {Lévy} Jump distribution functions. Phys. Rev. E 76, 041105, 8 pp.Google Scholar
Chakrabarty, A. and Meerschaert, M. M. (2011). Tempered stable laws as random walk limits. Statist. Prob. Lett. 81, 989–997.Google Scholar
Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976). A method for simulating stable random variables. J. Amer. Statist. Assoc. 71, 340–344.Google Scholar
Cohen, S. and Rosiński, J. (2007). Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes. Bernoulli 13, 195–210.Google Scholar
Cohen, S., Meerschaert, M. M. and Rosiński, J. (2010). Modeling and simulation with operator scaling. Stoch. Process. Appl. 120, 2390–2411.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principles Math. Sci. 288), 2nd edn. Springer, Berlin.Google Scholar
Jurek, Z. J. and Mason, J. D. (1993). Operator-Limit Distributions in Probability Theory. John Wiley, New York.Google Scholar
Koponen, I. (1995). Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E 52, 1197–1199.CrossRefGoogle Scholar
Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors. John Wiley, New York.Google Scholar
Meerschaert, M. M., Zhang, Y. and Baeumer, B. (2008). Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, 8–11.Google Scholar
Novikov, E. A. (1994). Infinitely divisible distributions in turbulence. Phys. Rev. E 50, R3303–R3305.CrossRefGoogle ScholarPubMed
Rosiński, J. (2007). Tempering stable processes. Stoch. Process. Appl. 117, 677–707.CrossRefGoogle Scholar
Zhang, Y., Baeumer, B. and Reeves, D. M. (2010). A tempered multiscaling stable model to simulate transport in regional-scale fractured media. Geophys. Res. Lett. 37, L11405, 5 pp.Google Scholar
Zhang, Y. et al. (2006). Random walk approximation of fractional-order multiscaling anomalous diffusion. Phys. Rev. E 74, 026706, 10 pp.Google Scholar