Exponential Ergodicity and β-Mixing Property for Generalized Ornstein-Uhlenbeck Processes

Author(s)
Oesook Lee

ABSTRACT

The generalized Ornstein-Uhlenbeck process is derived from a bivariate Lévy process and is suggested as a continuous time version of a stochastic recurrence equation [1]. In this paper we consider the generalized Ornstein-Uhlenbeck process and provide sufficient conditions under which the process is exponentially ergodic and hence holds the expo-nentially β-mixing property. Our results can cover a wide variety of areas by selecting suitable Lévy processes and be used as fundamental tools for statistical analysis concerning the processes. Well known stochastic volatility models in finance such as Lévy-driven Ornstein-Uhlenbeck process is examined as a special case.

The generalized Ornstein-Uhlenbeck process is derived from a bivariate Lévy process and is suggested as a continuous time version of a stochastic recurrence equation [1]. In this paper we consider the generalized Ornstein-Uhlenbeck process and provide sufficient conditions under which the process is exponentially ergodic and hence holds the expo-nentially β-mixing property. Our results can cover a wide variety of areas by selecting suitable Lévy processes and be used as fundamental tools for statistical analysis concerning the processes. Well known stochastic volatility models in finance such as Lévy-driven Ornstein-Uhlenbeck process is examined as a special case.

Cite this paper

O. Lee, "Exponential Ergodicity and β-Mixing Property for Generalized Ornstein-Uhlenbeck Processes,"*Theoretical Economics Letters*, Vol. 2 No. 1, 2012, pp. 21-25. doi: 10.4236/tel.2012.21004.

O. Lee, "Exponential Ergodicity and β-Mixing Property for Generalized Ornstein-Uhlenbeck Processes,"

References

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[2] D. B. Nelson, “ARCH Models as Diffusion Approximations,” Journal of Econometrics, Vol. 45, No. 1-2, 1990, pp. 7-38. doi:10.1016/0304-4076(90)90092-8

[3] C. Klüppelberg, A. Lindner and R. A. Maller, “A Continuous Time GARCH Process Driven by a Levy Process: Stationarity and Second Order Behavior,” Journal of Applied Probability, Vol. 41, No. 3, 2004, pp. 601-622. doi:10.1239/jap/1091543413

[4] O. E. Barndorff-Nielsen and N. Shephard, “Non-Gaussian Ornstein-Uhlenbeck Based Models and Some of Their Uses in Financial Economics (with dis-cussion),” Journal of Royal Statistical Society, Series B, Vol. 63, No. 2, 2001, pp. 167-241. doi:10.1111/1467-9868.00282

[5] R. A. Maller, G. Müller and A. Szimayer, “GARCH Modeling in Continuous Time for Irregularly Spaced Time Series Data,” Bernoulli, Vol. 14, No. 2, 2008, pp. 519-542. doi:10.3150/07-BEJ6189

[6] A. Lindner and R. A. Maller, “Levy Integrals and the Stationarity of Generalized Ornstein-Uhlenbeck Processes,” Stochastic Processes and Their Applications, Vol. 115, No. 10, 2005, pp. 1701-1722. doi:10.1016/j.spa.2005.05.004

[7] V. Fasen, “Asymptotic Results for Sample Auto-covariance Functions and Extremes of Integrated Generalized Ornstein-Uhlenbeck Processes,” 2010. http://www.ma.tum.de/stat/

[8] H. Masuda, “On Multidimen-sional Ornstein-Uhlenbeck Processes Driven by a General Levy Process,” Bernoulli, Vol. 10, No. 1, 2004, pp. 97-120. doi:10.3150/bj/1077544605

[9] C. Klüppelberg, A. Lindner and R. A. Maller, “Continuous Time Volatility Modeling: COGARCH versus Ornstein-Uhlenbeck Models,” In: Y. Ka-banov, R. Liptser, and J. Stoyanov, Eds., Stochastic Calculus to Mathematical Finance, The Shiryaev Festschrift, Springer, Berlin, 2006, pp. 393-419. doi:10.1007/978-3-540-30788-4_21

[10] A. Lindner, “Continuous Time Approximations to GARCH and Stochastic Volatility Models,” In: T. G. Andersen, R. A. Davis, J. P. Krei and T. Mikosch, Eds., Handbook of Financial Time Series, Springer, Berlin, 2010.

[11] S. P. Meyn and R. L. Tweedie, “Markov Chain and Stochastic Stability,” Springer-Verlag, Berlin, 1993.

[12] J. Bertoin, “Lévy Processes,” Cambridge University Press, Cambridge, 1996.

[13] K. Sato, “Levy Processes and Infinitely Divisible Distributions,” Cambridge University Press, Cambridge, 1999.

[14] P. Tuominen and R. L. Tweedie, “Exponential Decay and Ergodicity of General Markov Processes and Their Discrete Skeletons,” Advances in Applied Probability, Vol. 11, 1979, pp.784-803. doi:10.2307/1426859

[15] G. H. Hardy, J. E. Littlewood and G. Pólya, “Inequalities,” Cambridge University Press, Cambridge, 1952.

[16] J. Liu and E. Susko, “On Strict Stationarity and Ergodicity of a Nonlinear ARMA Model,” Journal of Applied Probability, Vol. 29, 1992, pp. 363-373. doi:10.2307/3214573

[17] R. L. Tweedie, “Drift Conditions and Invariant Measures for Markov Chains,” Stochastic Processes and their Applications, Vol. 92, No. 2, 2001, pp. 345-354. doi:10.1016/S0304-4149(00)00085-5

[18] K. Sato and M. Yamazato, “Operator-self Decomposable Distributions as Limit Distributions of Processes of Ornstein-Uhlenbeck Type,” Stochastic Processes and Their Applications, Vol. 17, No. 1, 1984, pp. 73-100. doi:10.1016/0304-4149(84)90312-0

[1] L. de Haan and R. L. Karandikar, “Embedding a Stochastic Difference Equation in a Continuous Time Process,” Stochastic Processes and Their Applications, Vol. 32, No. 2, 1989, pp. 225-235. doi:10.1016/0304-4149(89)90077-X

[2] D. B. Nelson, “ARCH Models as Diffusion Approximations,” Journal of Econometrics, Vol. 45, No. 1-2, 1990, pp. 7-38. doi:10.1016/0304-4076(90)90092-8

[3] C. Klüppelberg, A. Lindner and R. A. Maller, “A Continuous Time GARCH Process Driven by a Levy Process: Stationarity and Second Order Behavior,” Journal of Applied Probability, Vol. 41, No. 3, 2004, pp. 601-622. doi:10.1239/jap/1091543413

[4] O. E. Barndorff-Nielsen and N. Shephard, “Non-Gaussian Ornstein-Uhlenbeck Based Models and Some of Their Uses in Financial Economics (with dis-cussion),” Journal of Royal Statistical Society, Series B, Vol. 63, No. 2, 2001, pp. 167-241. doi:10.1111/1467-9868.00282

[5] R. A. Maller, G. Müller and A. Szimayer, “GARCH Modeling in Continuous Time for Irregularly Spaced Time Series Data,” Bernoulli, Vol. 14, No. 2, 2008, pp. 519-542. doi:10.3150/07-BEJ6189

[6] A. Lindner and R. A. Maller, “Levy Integrals and the Stationarity of Generalized Ornstein-Uhlenbeck Processes,” Stochastic Processes and Their Applications, Vol. 115, No. 10, 2005, pp. 1701-1722. doi:10.1016/j.spa.2005.05.004

[7] V. Fasen, “Asymptotic Results for Sample Auto-covariance Functions and Extremes of Integrated Generalized Ornstein-Uhlenbeck Processes,” 2010. http://www.ma.tum.de/stat/

[8] H. Masuda, “On Multidimen-sional Ornstein-Uhlenbeck Processes Driven by a General Levy Process,” Bernoulli, Vol. 10, No. 1, 2004, pp. 97-120. doi:10.3150/bj/1077544605

[9] C. Klüppelberg, A. Lindner and R. A. Maller, “Continuous Time Volatility Modeling: COGARCH versus Ornstein-Uhlenbeck Models,” In: Y. Ka-banov, R. Liptser, and J. Stoyanov, Eds., Stochastic Calculus to Mathematical Finance, The Shiryaev Festschrift, Springer, Berlin, 2006, pp. 393-419. doi:10.1007/978-3-540-30788-4_21

[10] A. Lindner, “Continuous Time Approximations to GARCH and Stochastic Volatility Models,” In: T. G. Andersen, R. A. Davis, J. P. Krei and T. Mikosch, Eds., Handbook of Financial Time Series, Springer, Berlin, 2010.

[11] S. P. Meyn and R. L. Tweedie, “Markov Chain and Stochastic Stability,” Springer-Verlag, Berlin, 1993.

[12] J. Bertoin, “Lévy Processes,” Cambridge University Press, Cambridge, 1996.

[13] K. Sato, “Levy Processes and Infinitely Divisible Distributions,” Cambridge University Press, Cambridge, 1999.

[14] P. Tuominen and R. L. Tweedie, “Exponential Decay and Ergodicity of General Markov Processes and Their Discrete Skeletons,” Advances in Applied Probability, Vol. 11, 1979, pp.784-803. doi:10.2307/1426859

[15] G. H. Hardy, J. E. Littlewood and G. Pólya, “Inequalities,” Cambridge University Press, Cambridge, 1952.

[16] J. Liu and E. Susko, “On Strict Stationarity and Ergodicity of a Nonlinear ARMA Model,” Journal of Applied Probability, Vol. 29, 1992, pp. 363-373. doi:10.2307/3214573

[17] R. L. Tweedie, “Drift Conditions and Invariant Measures for Markov Chains,” Stochastic Processes and their Applications, Vol. 92, No. 2, 2001, pp. 345-354. doi:10.1016/S0304-4149(00)00085-5

[18] K. Sato and M. Yamazato, “Operator-self Decomposable Distributions as Limit Distributions of Processes of Ornstein-Uhlenbeck Type,” Stochastic Processes and Their Applications, Vol. 17, No. 1, 1984, pp. 73-100. doi:10.1016/0304-4149(84)90312-0