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Modeling and simulation with operator scaling

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  • Cohen, Serge
  • Meerschaert, Mark M.
  • Rosinski, Jan
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    Abstract

    Self-similar processes are useful models for natural systems that exhibit scaling. Operator scaling allows a different scale factor in each coordinate. This paper develops practical methods for modeling and simulation. A simulation method is developed for operator scaling Lévy processes, based on a series representation, along with a Gaussian approximation of the small jumps. Several examples are given to illustrate the range of practical applications. A complete characterization of symmetries in two dimensions is given, for any exponent and spectral measure, to inform the choice of these model parameters. The paper concludes with some extensions to general operator self-similar processes.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 120 (2010)
    Issue (Month): 12 (December)
    Pages: 2390-2411

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    Handle: RePEc:eee:spapps:v:120:y:2010:i:12:p:2390-2411

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    Related research

    Keywords: Lévy processes Gaussian approximation Shot noise series expansions Simulation Tempered stable processes Operator stable processes;

    References

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    1. Sato, Ken-iti, 1987. "Strictly operator-stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 22(2), pages 278-295, August.
    2. Loretan, Mico & Phillips, Peter C. B., 1994. "Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets," Journal of Empirical Finance, Elsevier, vol. 1(2), pages 211-248, January.
    3. Meerschaert, Mark M. & Xiao, Yimin, 2005. "Dimension results for sample paths of operator stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 55-75, January.
    4. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2008. "Triangular array limits for continuous time random walks," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1606-1633, September.
    5. Meerschaert, Mark M. & Alan Veeh, Jeery, 1995. "Symmetry groups in d-space," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 1-6, January.
    6. Meerschaert, Mark M. & Scalas, Enrico, 2006. "Coupled continuous time random walks in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 114-118.
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