Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Continuous Time Approximations to GARCH(1, 1)-Family Models and Their Limiting Properties

  • Lee, O. (Department of Statistics, Ewha Womans University)
  • Received : 2014.05.18
  • Accepted : 2014.06.24
  • Published : 2014.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

Various modified GARCH(1, 1) models have been found adequate in many applications. We are interested in their continuous time versions and limiting properties. We first define a stochastic integral that includes useful continuous time versions of modified GARCH(1, 1) processes and give sufficient conditions under which the process is exponentially ergodic and ${\beta}$-mixing. The central limit theorem for the process is also obtained.

Keywords

References

  1. Bhattacharya, R. N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 60, 185-201. https://doi.org/10.1007/BF00531822
  2. Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, 31, 307-327. https://doi.org/10.1016/0304-4076(86)90063-1
  3. Donati-Martin, C., Ghomrasni, R. and Yor, M. (2001). On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options, Revista Mathmatica Iberoamericana, 17, 179-193.
  4. Doukhan, P. (1994). Mixing: Properties and Examples, Springer, New York.
  5. Duan, J. C. (1997). Augmented GARCH(p, q) process and its diffusion limit, Journal of Econometrics, 79, 97-127. https://doi.org/10.1016/S0304-4076(97)00009-2
  6. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation, Econometrica, 50, 987-1008. https://doi.org/10.2307/1912773
  7. Fasen, V. (2010). Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein-Uhlenbeck processes, Bernoulli, 16, 51-79. https://doi.org/10.3150/08-BEJ174
  8. Kallsen, J. and Vesenmayer, B. (2009). COGARCH as a continuous time limit of GARCH(1, 1), Stochastic Processes and Their Applications, 119, 74-98. https://doi.org/10.1016/j.spa.2007.12.008
  9. Kluppelberg, C., Lindner, A. and Maller, R. A. (2004). A continuous time GARCH process driven by a Levy process: Stationarity and second order behavior, Journal of Applied Probability, 41, 601-622. https://doi.org/10.1239/jap/1091543413
  10. Kluppelberg, C., Lindner, A. and Maller, R. A. (2006). Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models, in Kabanov, Y., Liptser, R. and Stoyanov, J. (Eds.), Stochastic Calculus to Mathematical Finance, Springer, Berlin, 393-419.
  11. Lee, O. (2012a). V-uniform ergodicity of a continuous time asymmetric power GARCH(1, 1) model, Statistics and Probability Letters, 82, 812-817. https://doi.org/10.1016/j.spl.2012.01.006
  12. Lee, O. (2012b). Exponential ergodicity and ${\beta}$-mixing property for generalized Ornstein-Uhlenbeck processes, Theoretical Economics Letters, 2, 21-25. https://doi.org/10.4236/tel.2012.21004
  13. Lindner, A. (2009). Continuous time approximations to GARCH and stochastic volatility models, in Andersen, T.G., Davis, R.A., Kreiss, J.P. and Mikosch, T.(Eds.), Handbook of Financial Time Series, Springer, Berlin, 481-496.
  14. Maller, R. A., Muller, G. and Szimayer, A. (2008). GARCH modelling in continuous time for irregularly spaced time series data, Bernoulli, 14, 519-542. https://doi.org/10.3150/07-BEJ6189
  15. Meyn, S. P. and Tweedie, R. L. (1993). Markov Chain and Stochastic Stability, Springer-Verlag, Berlin.
  16. Nelson, D. B. (1990). ARCH models as diffusion approximations, Journal of Econometrics, 45, 7-38. https://doi.org/10.1016/0304-4076(90)90092-8
  17. Protter, P. (2005). Stochastic Integration and Differential Equations, Springer, New York.
  18. Sato, K. (1999). Levy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge.
  19. Tuominen, P. and Tweedie, R. L. (1979). Exponential decay and ergodicity of general Markov processes and their discrete skeletons, Advances in Applied Probability, 11, 784-803. https://doi.org/10.2307/1426859