Lévy–Driven Continuous–Time ARMA Processes

  • Peter J. BrockwellEmail author


Gaussian ARMA processes with continuous time parameter, otherwise known as stationary continuous-time Gaussian processes with rational spectral density, have been of interest for many years. (See for example the papers of Doob (1944), Bartlett (1946), Phillips (1959), Durbin (1961), Dzhapararidze (1970,1971), Pham-Din-Tuan (1977) and the monograph of Arató (1982).) In the last twenty years there has been a resurgence of interest in continuous-time processes, partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations (Hull and White (1987)). Numerous examples of econometric applications of continuous-time models are contained in the book of Bergstrom (1990). Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data (Jones (1981, 1985), Jones and Ackerson (1990)). Like their discrete-time counterparts, continuous-time ARMA processes constitute a very convenient parametric family of stationary processes exhibiting a wide range of autocorrelation functions which can be used to model the empirical autocorrelations observed in financial time series analysis. In financial applications it has been observed that jumps play an important role in the realistic modelling of asset prices and derived series such as volatility. This has led to an upsurge of interest in Lévy processes and their applications to financial modelling. In this article we discuss second-order Lévy-driven continuous-time ARMA models, their properties and some of their financial applications. Examples are the modelling of stochastic volatility in the class of models introduced by Barndorff-Nielsen and Shephard (2001) and the construction of a class of continuous-time GARCH models which generalize the COGARCH(1,1) process of Klüppelberg, Lindner and Maller (2004) and which exhibit properties analogous to those of the discretetime GARCH(p,q) process.


Stochastic Volatility Negative Real Part Stochastic Volatility Model Financial Time Series Volatility Process 
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  1. Applebaum, D. (2004): Lévy Processes and Stochastic Calculus Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  2. Arató, M. (1982): Linear Stochastic Systems with Constant Coefficients Springer Lecture Notes in Control and Information Systems 45, Springer-Verlag, Berlin.Google Scholar
  3. Barndorff-Nielsen, O.E. (2001): Superposition of Ornstein-Uhlenbeck type processes. Theory Probab. Appl. 45, 175–194.CrossRefMathSciNetGoogle Scholar
  4. 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. Roy. Statist. Soc. Ser. B 63, 167–241.CrossRefMathSciNetGoogle Scholar
  5. Barndorff-Nielsen, O.E., Nicolato E. and Shephard, N. (2002): Some recent developments in stochastic volatility modelling. Quantitative Finance 2, 11–23.CrossRefMathSciNetGoogle Scholar
  6. Bartlett, M.S. (1946): On the theoretical specification and sampling properties of autocorrelated time series. J. Royal Statistical Soc. (Supplement) 7, 27–41.Google Scholar
  7. Bergstrom, A.R. (1985): The estimation of parameters in non-stationary higher-order continuous-time dynamic models. Econometric Theory 1, 369–385.MathSciNetGoogle Scholar
  8. Bergstrom, A.R. (1990): Continuous Time Econometric Modelling Oxford University Press, Oxford.Google Scholar
  9. Bertoin, J. (1996): Lévy Processes Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  10. Brockwell, A.E. and Brockwell, P.J. (1998): A class of non-embeddable ARMA processes. J. Time Series Analysis 20, 483–486.MathSciNetGoogle Scholar
  11. Brockwell, P.J. (1995): A note on the embedding of discrete-time ARMA processes. J. Time Series Analysis 16, 451–460.CrossRefMathSciNetGoogle Scholar
  12. Brockwell, P.J. (2000): Continuous–time ARMA processes. In: C.R. Rao and D.N. Shanbhag (Eds.): Stochastic processes: theory and methods, Handbook of Statist. 19, 249–276. North–Holland, Amsterdam.Google Scholar
  13. Brockwell, P.J. (2001): Lévy-driven CARMA processes. Ann. Inst. Stat. Mat. 53, 113–124.MathSciNetGoogle Scholar
  14. Brockwell, P.J. (2004): Representations of continuous-time ARMA processes. J. Appl. Probab. 41A, 375–382.CrossRefMathSciNetGoogle Scholar
  15. Brockwell, P.J. and Davis, R.A. (1991): Time Series: Theory and Methods 2nd edition. Springer, New York.Google Scholar
  16. Brockwell, P.J. and Marquardt, T. (2005): Fractionally integrated continuous-time ARMA processes. Statistica Sinica 15, 477–494.zbMATHMathSciNetGoogle Scholar
  17. Brockwell, P.J., Chadraa, E., and Lindner, A. (2006): Continuous-time GARCH processes. Annals Appl. Prob. 16, 790–826.zbMATHCrossRefMathSciNetGoogle Scholar
  18. Brockwell, P.J., Davis, R.A. and Yang, Y. (2007): Continuous-time autoregression. Statistica Sinica 17, 63–80.zbMATHGoogle Scholar
  19. Brockwell, P.J., Davis, R.A. and Yang, Y. (2007): Inference for non-negative Lévy-driven Ornstein-Uhlenbeck processes. J. Appl. Prob. 44, 977–989.zbMATHCrossRefMathSciNetGoogle Scholar
  20. Chan, K.S. and Tong, H. (1987): A Note on embedding a discrete parameter ARMA model in a continuous parameter ARMA model. J. Time Ser. Anal. 8, 277–281.zbMATHCrossRefMathSciNetGoogle Scholar
  21. Doob, J.L. (1944): The elementary Gaussian processes. Ann. Math. Statist. 25, 229–282.CrossRefMathSciNetGoogle Scholar
  22. Durbin, J. (1961): Efficient fitting of linear models for continuous stationary time series from discrete data. Bull. Int. Statist. Inst. 38, 273–281.zbMATHMathSciNetGoogle Scholar
  23. Dzhaparidze, K.O. (1970): On the estimation of the spectral parameters of a stationary Gaussian process with rational spectral density. Th. Prob. Appl. 15, 531–538.CrossRefGoogle Scholar
  24. Dzhaparidze, K.O. (1971): On methods for obtaining asymptotically efficient spectral parameter estimates for a stationary Gaussian process with rational spectral density. Th. Prob. Appl. 16, 550–554.zbMATHCrossRefGoogle Scholar
  25. Eberlein, E. and Raible, S. (1999): Term structure models driven by general Lévy processes. Mathematical Finance 9, 31–53.zbMATHCrossRefMathSciNetGoogle Scholar
  26. Fasen, V. (2004): Lévy Driven MA Processes with Applications in Finance. Ph.D. thesis, Technical University of Munich.Google Scholar
  27. Hull, J. and White, A. (1987): The pricing of assets on options with stochastic volatilities. J. of Finance 42, 281–300.CrossRefGoogle Scholar
  28. Hyndman, R.J. (1993): Yule-Walker estimates for continuous-time autoregressive models. J. Time Series Analysis 14, 281–296.zbMATHCrossRefMathSciNetGoogle Scholar
  29. Jones, R.H. (1981): Fitting a continuous time autoregression to discrete data. In: Findley, D.F. (Ed.): Applied Time Series Analysis II, 651–682. Academic Press, New York.Google Scholar
  30. Jones, R.H. (1985): Time series analysis with unequally spaced data. In: Hannan, E.J., Krishnaiah, P.R. and Rao, M.M. (Eds.): Time Series in the Time Domain, Handbook of Statistics 5, 157–178. North Holland, Amsterdam.Google Scholar
  31. Jones, R.H. and Ackerson, L.M. (1990): Serial correlation in unequally spaced longitudinal data. Biometrika 77, 721–732.CrossRefMathSciNetGoogle Scholar
  32. Jongbloed, G., van der Meulen, F.H. and van der Vaart, A.W. (2005): Non-parametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli 11, 759–791.zbMATHCrossRefMathSciNetGoogle Scholar
  33. Klüppelberg, C., Lindner, A. and Maller, R. (2004): A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour. J. Appl. Probab. 41, 601–622.zbMATHCrossRefMathSciNetGoogle Scholar
  34. Lindner, A. (2008): 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, 481–496. Springer, New York.Google Scholar
  35. Pham-Din-Tuan (1977): Estimation of parameters of a continuous-time Gaussian stationary process with rational spectral density function. Biometrika 64, 385–399.CrossRefMathSciNetGoogle Scholar
  36. Phillips, A.W. (1959): The estimation of parameters in systems of stochastic differential equations. Biometrika 46, 67–76.zbMATHMathSciNetGoogle Scholar
  37. Protter, P.E. (2004): Stochastic Integration and Differential Equations. 2nd edition. Springer, New York.zbMATHGoogle Scholar
  38. Sato, K. (1999): Lévy Processes and Infinitely Divisible Distributions Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  39. Todorov, V. (2007): Econometric analysis of jump-driven stochastic volatility models.
  40. Todorov, V. and Tauchen, G. (2006): Simulation methods for Lévy-driven CARMA stochastic volatility models. J. Business and Economic Statistics 24, 455–469.CrossRefMathSciNetGoogle Scholar
  41. Tsai, H. and Chan, K.S. (2005): A note on non-negative continuous-time processes. J. Roy. Statist. Soc. Ser. B 67, 589–597.zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of StatisticsColorado State UniversityFort CollinsU.S.A.

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