Abstract
Properties and examples of continuous-time ARMA (CARMA) processes driven by Lévy processes are examined. By allowing Lévy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Lévy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Lévy processes to investigate CARMA processes driven by Lévy processes {W(t)} without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.
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References
Barndorff-Nielsen, O. E. and Shephard, N. (1999). Non-Gaussian OU based models and some of their uses in financial economics, Working Paper in Economics, Nuffield College, Oxford.
Bertoin, Jean (1996). Lévy Processes, Cambridge University Press, Cambridge.
Brockwell, P. J. (2000a). Continuous-time ARMA Processes, Handbook of Statistics: Stochastic Processes, Theory and Methods (eds. C. R. Rao and D. N. Shanbhag), Elsevier, Amsterdam.
Brockwell, P. J. (2000b). Heavy-tailed and non-linear continuous-time ARMA models for financial time series, Statistics and Finance: An Interface (eds. W. S. Chan, W. K. Li and H. Tong), Imperial College Press, London.
Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting, Springer, New York.
Brockwell, P. J. and Williams, R. J. (1997). On the existence and application of continuous-time threshold autoregressions of order two, Advances in Applied Probability, 29, 205-227.
Brockwell, P. J., Resnick, S. I. and Tweedie, R. L. (1982). Storage processes with general release rule and additive inputs, Advances in Applied Probability, 14, 392-433.
Cinlar, E. and Pinsky, M. (1972). On dams with additive inputs and a general release rule, J. Appl. Probab, 9, 422-429.
Granger, C. W., Ding, Z. and Spear, S. (1999). Stylized facts on the temporal and distributional properties of absolute returns; an update, Paper presented at Hong Kong International Workshop on Statistics in Finance.
Harrison, J. M. and Resnick, S. I. (1976). The stationary distribution and first exit probabilities of a storage process with general release rule, Math. Oper. Res., 1, 347-358.
Ito, K. (1969). Stochastic Processes, Lecture Note Series, 16, Matematisk Institut, Aarhus University.
Jones, R. H. (1981). Fitting a continuous time autoregression to discrete data, Applied Time Series Analysis II (ed. D. F. Findley), 651-682, Academic Press, New York.
Jones, R. H. (1985). Time series analysis with unequally spaced data, Time Series in the Time Domain, Handbook of Statistics, 5 (eds. E. J. Hannan, P. R. Krishnaiah and M. M. Rao), 157-178, North Holland, Amsterdam.
Küchler, U. and Sørensen, M. (1997). Exponential Families of Stochastic Processes, Springer, New York.
Ozaki, T. (1985). Non-linear time series models and dynamical systems, Time Series in the Time Domain, Handbook of Statistics, 5 (eds. E. J. Hannan, P. R. Krishnaiah and M. M. Rao), 25-84, North Holland, Amsterdam.
Protter, P. (1991). Stochastic Integration and Differential Equations, A New Approach, Springer, New York.
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Brockwell, P.J. Lévy-Driven Carma Processes. Annals of the Institute of Statistical Mathematics 53, 113–124 (2001). https://doi.org/10.1023/A:1017972605872
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DOI: https://doi.org/10.1023/A:1017972605872