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A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

Part of: Markov processes Mathematical economics Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Claudia Klüppelberg ,
Alexander Lindner  and
Ross Maller
Claudia Klüppelberg*
Affiliation:
Munich University of Technology
Alexander Lindner*
Affiliation:
Munich University of Technology
Ross Maller*
Affiliation:
Australian National University
*
∗ Postal address: Center for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany
∗ Postal address: Center for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany
∗∗∗∗ Postal address: Centre for Mathematical Analysis and School of Finance and Applied Statistics, Australian National University, Canberra, ACT 0200, Australia. Email address: ross.maller@anu.edu.au
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Abstract

We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.

Keywords

ARCH modelGARCH modelstabilitystationarityconditional heteroscedasticityperpetuitiesstochastic integrationLévy process

MSC classification

Primary: 60G10: Stationary processes 60J25: Continuous-time Markov processes on general state spaces 91B70: Stochastic models
Secondary: 91B84: Economic time series analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 601 - 622
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, 730–747.10.1239/jap/1037816015Google Scholar
Barndorff-Nielsen, O. E., and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Statist. Soc. B 63, 167–241.10.1111/1467-9868.00282Google Scholar
Barndorff-Nielsen, O. E., and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes, Theory and Applications, eds Barndorff-Nielsen, O. E., Mikosch, T. and Resnick, S., Birkhäuser, Boston, pp. 283–318.10.1007/978-1-4612-0197-7_13Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J., and Yor, M. (2002). On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse Math. (6) 11, 33–45.10.5802/afst.1016Google Scholar
Bollerslev, T., Engle, R. F., and Nelson, D. B. (1995). ARCH models. In The Handbook of Econometrics, Vol. 4, eds Engle, R. F. and McFadden, D., North-Holland, Amsterdam, pp. 2959–3038.10.1016/S1573-4412(05)80018-2Google Scholar
Bougerol, P., and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52, 115–127.10.1016/0304-4076(92)90067-2Google Scholar
De Haan, L., and Karandikar, R. L. (1989). Embedding a stochastic difference equation in a continuous-time process. Stoch. Process. Appl. 32, 225–235.10.1016/0304-4149(89)90077-XGoogle Scholar
Drost, F. C., and Werker, B. J. M. (1996). Closing the GARCH gap: continuous time GARCH modelling. J. Econometrics 74, 31–57.10.1016/0304-4076(95)01750-XGoogle Scholar
Duan, J.-C. (1997). Augmented GARCH(p,q) process and its diffusion limit. J. Econometrics 79, 97–127.10.1016/S0304-4076(97)00009-2Google Scholar
Eberlein, E. (2001). Application of generalised hyperbolic Lévy motions to finance. In Lévy Processes, Theory and Applications, eds Barndorff-Nielsen, O. E., Mikosch, T. and Resnick, S., Birkhäuser, Boston, pp. 319–336.10.1007/978-1-4612-0197-7_14Google Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.10.1007/978-3-642-33483-2Google Scholar
Engle, R. F. (1995). ARCH: Selected Readings. Oxford University Press.Google Scholar
Erickson, K. B. (1973). The strong law of large numbers when the mean is undefined. Trans. Amer. Math. Soc. 185, 371–381.10.1090/S0002-9947-1973-0336806-5Google Scholar
Erickson, K. B., and Maller, R. A. (2004). Generalised Ornstein-Uhlenbeck processes and the convergence of Lévy integrals. To appear in Sminaire de Probabilit's (Lecture Notes Math.), Springer, Berlin.Google Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.10.1214/aoap/1177005985Google Scholar
Goldie, C. M., and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob. 28, 1195–1218.Google Scholar
Kesten, H., and Maller, R. (1996). Two renewal theorems for general random walks tending to infinity. Prob. Theory Relat. Fields 106, 1–38.10.1007/s004400050056Google Scholar
Madan, D., and Seneta, E. (1990). The variance gamma (VG) model for share market returns, J. Business 63, 511–524.10.1086/296519Google Scholar
Mikosch, T. and Stărică, C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process. Ann. Statist. 28, 1427–1451.Google Scholar
Nelson, D. B. (1990). ARCH models as diffusion approximations. J. Econometrics 45, 7–38.10.1016/0304-4076(90)90092-8Google Scholar
Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econom. Theory 6, 318–334.10.1017/S0266466600005296Google Scholar
Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, New York.Google Scholar
Rogers, L. C. G., and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus, 2nd edn. Cambridge University Press.Google Scholar
Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Processes. Chapman and Hall, London.Google Scholar
Sampson, M. (1988). A stationarity condition for the GARCH(1,1) process. Mimeo, Department of Economics, Concordia University, Montreal.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Schoutens, W., and Teugels, J. L. (1998). Lévy processes, polynomials and martingales. Commun. Statist. Stoch. Models 14, 335–349.10.1080/15326349808807475Google Scholar
Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. In Time Series Models in Econometrics, Finance and other Fields, eds Barndorff-Nielsen, O. E., Cox, D. R. and Hinkley, D. V., Chapman and Hall, London, pp. 1–67.Google Scholar