Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
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A continuous time asymmetric power GARCH process driven by a L$\'{e}$vy process

  • Lee, Oe-Sook (Department of Statistics, Ewha Womans University)
  • Received : 2010.09.11
  • Accepted : 2010.11.05
  • Published : 2010.11.30
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Abstract

A continuous time asymmetric power GARCH(1,1) model is suggested, based on a single background driving L$\'{e}$vy process. The stochastic differential equation for the given process is derived and the strict stationarity and kth order moment conditions are examined.

Keywords

References

  1. Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics. Journal of Royal Statistical Society Series B, 63, 167-241. https://doi.org/10.1111/1467-9868.00282
  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. Brockwell, P. (2001). Levy driven CARMA process. Annals Institute Statistical Mathematics, 53, 113-124. https://doi.org/10.1023/A:1017972605872
  4. Ding, Z., Granger, C. W. J. and Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1, 83-106. https://doi.org/10.1016/0927-5398(93)90006-D
  5. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica, 50, 987-1008. https://doi.org/10.2307/1912773
  6. Gordie, C. M. and Maller, R. A. (2000). Stability of perpetuities. The Annals of Probability, 28, 1195-1218. https://doi.org/10.1214/aop/1019160331
  7. Haug, S. and Czado, C. (2007). An exponential continuous time GARCH process. Journal of Applied Probability, 44, 960-976. https://doi.org/10.1239/jap/1197908817
  8. Haug, S., Kluppelberg, C., Lindner, A. and Zapp, M. (2007). Method of moment estimation in the COGARCH(1,1) model. The Econometrics Journal, 10, 320-341. https://doi.org/10.1111/j.1368-423X.2007.00210.x
  9. Hentschel, L. (1995). All in the family nesting symmetric and asymmetric GARCH models. Journal of Financial Economics, 39, 71-104. https://doi.org/10.1016/0304-405X(94)00821-H
  10. 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
  11. Kluppelberg, C., Lindner, A. and Maller, R. (2004). A continuous time GARCH process driven by a Levy process: Stationarity and second order behavior. Journal Applied Probability, 41, 601-622. https://doi.org/10.1239/jap/1091543413
  12. Kluppelberg, C., Lindner, A. and Maller, R. (2006). Continuous time volatility modelling: COGARCH versus Ornstein Uhlenbeck models: In Kabanov, Y., Lipster, R. and Stoyanov, J. (Eds.) from Stochastic calculus to mathematical finance, 393-419, Berlin, Springer.
  13. Lee, O. and Kim, M. J. (2006). Long memory and covariance stationarity of asymmetric power FIGARCH model. Journal of the Korean Data & Information Science Society, 17, 983-990.
  14. Maller, R., 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. 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
  16. Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347-370. https://doi.org/10.2307/2938260
  17. Park, S. Y. and Lee, S. Y. (2007). Modelling KOSPI200 data based on GARCH(1.1) parameter change test. Journal of the Korean Data & Information Science Society, 18, 11-16.
  18. Protter, P. E. (2004). Stochastic integration and differential equations 2nd edition, Springer, New York.