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Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

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  • Reiichiro Kawai

    (University of Sydney)

Abstract

We address the issue of sample path simulation of Lévy-driven continuous-time autoregressive moving average (CARMA) processes. Approximate discrete-time simulation schemes are constructed along with quantifiable error analysis for stable, second-order and non-negative CARMA processes, based upon the so-called series representation of infinitely divisible laws and associated Lévy processes. We prove that under suitable conditions, the simulation scheme can be improved in terms of second-order structure, finite dimensional laws as well as sample path properties. The simulation procedure is often quite simple and allows one to conduct super-sampling without running the algorithm once again. The computational complexity of the proposed scheme is not affected much by the sampling scheme, such as sampling frequency and irregular spacing. Numerical results are presented throughout to illustrate the effectiveness of the proposed simulation scheme.

Suggested Citation

  • Reiichiro Kawai, 2017. "Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 175-211, March.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:1:d:10.1007_s11009-015-9472-5
    DOI: 10.1007/s11009-015-9472-5
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    References listed on IDEAS

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    1. Shibin Zhang & Xinsheng Zhang, 2008. "Exact Simulation of IG-OU Processes," Methodology and Computing in Applied Probability, Springer, vol. 10(3), pages 337-355, September.
    2. Imai, Junichi & Kawai, Reiichiro, 2011. "On finite truncation of infinite shot noise series representation of tempered stable laws," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4411-4425.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.
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    1. Kawai, Reiichiro, 2021. "A general approach to sample path generation of infinitely divisible processes via shot noise representation," Statistics & Probability Letters, Elsevier, vol. 174(C).

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