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Spectral Representation of Multivariate Regularly Varying Lévy and CARMA Processes

Author

Listed:
  • Florian Fuchs

    (Technische Universität München)

  • Robert Stelzer

    (Ulm University)

Abstract

A spectral representation for regularly varying Lévy processes with index between one and two is established and the properties of the resulting random noise are discussed in detail, giving also new insight in the L 2-case where the noise is a random orthogonal measure. This allows a spectral definition of multivariate regularly varying Lévy-driven continuous time autoregressive moving average (CARMA) processes. It is shown that they extend the well-studied case with finite second moments and coincide with definitions previously used in the infinite variance case when they apply.

Suggested Citation

  • Florian Fuchs & Robert Stelzer, 2013. "Spectral Representation of Multivariate Regularly Varying Lévy and CARMA Processes," Journal of Theoretical Probability, Springer, vol. 26(2), pages 410-436, June.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:2:d:10.1007_s10959-011-0369-0
    DOI: 10.1007/s10959-011-0369-0
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    References listed on IDEAS

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    1. Brockwell, Peter J. & Lindner, Alexander, 2009. "Existence and uniqueness of stationary Lévy-driven CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2660-2681, August.
    2. P. Brockwell, 2001. "Lévy-Driven Carma Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(1), pages 113-124, March.
    3. Marquardt, Tina & Stelzer, Robert, 2007. "Multivariate CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 96-120, January.
    4. Marquardt, Tina, 2007. "Multivariate fractionally integrated CARMA processes," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1705-1725, October.
    5. Todorov, Viktor & Tauchen, George, 2006. "Simulation Methods for Levy-Driven Continuous-Time Autoregressive Moving Average (CARMA) Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 455-469, October.
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