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Representations of continuous-time ARMA processes

Part of: Applications Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Peter J. Brockwell
Peter J. Brockwell*
Affiliation:
Department of Statistics, Colorado State University, Fort Collins, CO 80523-1877, USA. Email address: [email protected]
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Abstract

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.

Keywords

Continuous-time ARMA processLévy processstochastic volatilitylong memoryfractional integration

MSC classification

Primary: 60G10: Stationary processes
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
Part 7. Time series analysis
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 375 - 382
Copyright
Copyright © Applied Probability Trust 2004 

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References

Anh, V. V., Heyde, C. C. and Leonenko, N. N. (2002). Dynamic models of long-memory processes driven by Lévy noise. J. Appl. Prob. 39, 730–747.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B 63, 167–241.Google Scholar
Brockwell, P. J. (2000). Heavy-tailed and non-linear continuous-time ARMA models for financial time series. In Statistics and Finance: An Interface , eds Chan, W. S., Li, W. K. and Tong, H., Imperial College Press, London, pp. 3–22.Google Scholar
Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53, 113–124.Google Scholar
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. Springer, New York.Google Scholar
Comte, F. and Renault, E. (1996). Long memory continuous time models. J. Econometrics 73, 101–149.CrossRefGoogle Scholar
Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transform. Springer, Berlin.Google Scholar