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CARMA processes as solutions of integral equations

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  • Brockwell, Peter J.
  • Lindner, Alexander

Abstract

A CARMA(p,q) process is defined by suitable interpretation of the formal pth order differential equation a(D)Yt=b(D)DLt, where L is a two-sided Lévy process, a(z) and b(z) are polynomials of degrees p and q, respectively, with p>q, and D denotes the differentiation operator. Since derivatives of Lévy processes do not exist in the usual sense, the rigorous definition of a CARMA process is based on a corresponding state space equation. In this note, we show that the state space definition is also equivalent to the integral equation a(D)JpYt=b(D)Jp−1Lt+rt, where J, defined by Jft:=∫0tfsds, denotes the integration operator and rt is a suitable polynomial of degree at most p−1. This equation has well defined solutions and provides a natural interpretation of the formal equation a(D)Yt=b(D)DLt.

Suggested Citation

  • Brockwell, Peter J. & Lindner, Alexander, 2015. "CARMA processes as solutions of integral equations," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 221-227.
    • Handle: RePEc:eee:stapro:v:107:y:2015:i:c:p:221-227
    DOI: 10.1016/j.spl.2015.08.026
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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. Peter Brockwell & Alexander Lindner, 2013. "Integration of CARMA processes and spot volatility modelling," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(2), pages 156-167, March.
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    4. Tucker McElroy, 2013. "Forecasting continuous-time processes with applications to signal extraction," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(3), pages 439-456, June.
    5. Brockwell, Peter J. & Lindner, Alexander, 2015. "Prediction of Lévy-driven CARMA processes," Journal of Econometrics, Elsevier, vol. 189(2), pages 263-271.
    6. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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