Lévy-Driven Carma Processes

  • P. J. Brockwell
Article

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.

Lévy process CARMA process stochastic differential equation stable process 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.Google Scholar
  2. Bertoin, Jean (1996). Lévy Processes, Cambridge University Press, Cambridge.Google Scholar
  3. 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.Google Scholar
  4. 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.Google Scholar
  5. Brockwell, P. J. and Davis, R. A. (1996). Introduction to Time Series and Forecasting, Springer, New York.Google Scholar
  6. 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.Google Scholar
  7. 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.Google Scholar
  8. Cinlar, E. and Pinsky, M. (1972). On dams with additive inputs and a general release rule, J. Appl. Probab, 9, 422-429.Google Scholar
  9. 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.Google Scholar
  10. 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.Google Scholar
  11. Ito, K. (1969). Stochastic Processes, Lecture Note Series, 16, Matematisk Institut, Aarhus University.Google Scholar
  12. 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.Google Scholar
  13. 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.Google Scholar
  14. Küchler, U. and Sørensen, M. (1997). Exponential Families of Stochastic Processes, Springer, New York.Google Scholar
  15. 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.Google Scholar
  16. Protter, P. (1991). Stochastic Integration and Differential Equations, A New Approach, Springer, New York.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 2001

Authors and Affiliations

  • P. J. Brockwell
    • 1
  1. 1.Statistics DepartmentColorado State UniversityFort CollinsUSA

Personalised recommendations