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Multivariate stochastic delay differential equations and CAR representations of CARMA processes

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  • Basse-O’Connor, Andreas
  • Nielsen, Mikkel Slot
  • Pedersen, Jan
  • Rohde, Victor

Abstract

In this study we show how to represent a continuous time autoregressive moving average (CARMA) as a higher order stochastic delay differential equation, which may be thought of as a CAR(∞) representation. Furthermore, we show how the CAR(∞) representation gives rise to a prediction formula for CARMA processes. To be used in the above mentioned results we develop a general theory for multivariate stochastic delay differential equations, which will be of independent interest, and which will have particular focus on existence, uniqueness and representations.

Suggested Citation

  • Basse-O’Connor, Andreas & Nielsen, Mikkel Slot & Pedersen, Jan & Rohde, Victor, 2019. "Multivariate stochastic delay differential equations and CAR representations of CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4119-4143.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:10:p:4119-4143
    DOI: 10.1016/j.spa.2018.11.011
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    References listed on IDEAS

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    1. Brockwell, Peter J. & Davis, Richard A. & Yang, Yu, 2011. "Estimation for Non-Negative Lévy-Driven CARMA Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(2), pages 250-259.
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    4. Marquardt, Tina & Stelzer, Robert, 2007. "Multivariate CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 96-120, January.
    5. 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.
    6. 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.
    7. Fred Espen Benth & Jūratė Šaltytė Benth & Steen Koekebakker, 2007. "Putting a Price on Temperature," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 746-767, December.
    8. Marquardt, Tina, 2007. "Multivariate fractionally integrated CARMA processes," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1705-1725, October.
    9. Todorov, Viktor, 2009. "Estimation of continuous-time stochastic volatility models with jumps using high-frequency data," Journal of Econometrics, Elsevier, vol. 148(2), pages 131-148, February.
    10. Brockwell, Peter J. & Lindner, Alexander, 2015. "Prediction of Lévy-driven CARMA processes," Journal of Econometrics, Elsevier, vol. 189(2), pages 263-271.
    11. Gushchin, Alexander A. & Küchler, Uwe, 2000. "On stationary solutions of delay differential equations driven by a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 195-211, August.
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    Cited by:

    1. Nielsen, Mikkel Slot, 2020. "On non-stationary solutions to MSDDEs: Representations and the cointegration space," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3154-3173.

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