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Existence and uniqueness of stationary Lévy-driven CARMA processes

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  • Brockwell, Peter J.
  • Lindner, Alexander

Abstract

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining a general Lévy-driven continuous-parameter ARMA process with index set are determined. Under these conditions the solution is shown to be unique and an explicit expression is given for the process as an integral with respect to the background driving Lévy process. The results generalize results obtained earlier for second-order processes and for processes defined by the Ornstein-Uhlenbeck equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:8:p:2660-2681
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    References listed on IDEAS

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    1. 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.
    2. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
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    Citations

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    Cited by:

    1. Pham, Viet Son, 2020. "Lévy-driven causal CARMA random fields," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7547-7574.
    2. Fasen, Vicky & Fuchs, Florian, 2013. "On the limit behavior of the periodogram of high-frequency sampled stable CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 229-273.
    3. Spangenberg, Felix, 2013. "Strictly stationary solutions of ARMA equations in Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 127-138.
    4. Brockwell, Peter J. & Schlemm, Eckhard, 2013. "Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 217-251.
    5. Wang, Fangfang & Ma, Chunsheng, 2019. "ℓ1-symmetric vector random fields," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2466-2484.
    6. Peter J. Brockwell & Yasumasa Matsuda, 2017. "Continuous auto-regressive moving average random fields on ℝ-super-n," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 833-857, June.
    7. Vicky Fasen & Florian Fuchs, 2013. "Spectral estimates for high-frequency sampled continuous-time autoregressive moving average processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(5), pages 532-551, September.
    8. Appleby, John A.D. & Patterson, Denis D., 2021. "Growth and fluctuation in perturbed nonlinear Volterra equations," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    9. Brockwell, Peter J. & Lindner, Alexander, 2015. "CARMA processes as solutions of integral equations," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 221-227.
    10. Benth, Fred Espen & Taib, Che Mohd Imran Che, 2013. "On the speed towards the mean for continuous time autoregressive moving average processes with applications to energy markets," Energy Economics, Elsevier, vol. 40(C), pages 259-268.
    11. 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.
    12. Ernst, Philip A. & Brown, Lawrence D. & Shepp, Larry & Wolpert, Robert L., 2017. "Stationary Gaussian Markov processes as limits of stationary autoregressive time series," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 180-186.
    13. Brockwell, Peter J. & Lindner, Alexander, 2015. "Prediction of Lévy-driven CARMA processes," Journal of Econometrics, Elsevier, vol. 189(2), pages 263-271.
    14. Berger, David, 2020. "Lévy driven CARMA generalized processes and stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 5865-5887.
    15. Peter J. Brockwell & Yasumasa Matsuda, 2015. "Levy-driven CARMA Random Fields on Rn," TERG Discussion Papers 339, Graduate School of Economics and Management, Tohoku University.
    16. Ragnhild Noven & Almut Veraart & Axel Gandy, 2015. "A Lévy-driven rainfall model with applications to futures pricing," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 99(4), pages 403-432, October.
    17. Benth, Fred Espen & Klüppelberg, Claudia & Müller, Gernot & Vos, Linda, 2014. "Futures pricing in electricity markets based on stable CARMA spot models," Energy Economics, Elsevier, vol. 44(C), pages 392-406.

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