• Bernoulli
  • Volume 16, Number 1 (2010), 80-115.

Multivariate COGARCH(1, 1) processes

Robert Stelzer

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Multivariate COGARCH(1, 1) processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent time-varying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices.

After defining the COGARCH(1, 1) process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving Lévy process, explicit expressions for the first and second-order moments and (asymptotic) second-order stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and second-order structure of the increments of the multivariate COGARCH(1, 1) process and their “squares”.

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Bernoulli, Volume 16, Number 1 (2010), 80-115.

First available in Project Euclid: 12 February 2010

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COGARCH Lévy process multivariate GARCH positive definite random matrix process second-order moment structure stationarity stochastic differential equations stochastic volatility variance mixture model


Stelzer, Robert. Multivariate COGARCH(1, 1) processes. Bernoulli 16 (2010), no. 1, 80--115. doi:10.3150/09-BEJ196.

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  • [1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 93. Cambridge: Cambridge Univ. Press.
  • [2] Barndorff-Nielsen, O.E. and Pérez-Abreu, V. (2003). Extensions of type G and marginal infinite divisibility. Theory Probab. Appl. 47 202–218.
  • [3] Barndorff-Nielsen, O.E. and Pérez-Abreu, V. (2008). Matrix subordinators and related Upsilon transformations. Theory Probab. Appl. 52 1–23.
  • [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. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241.
  • [5] Barndorff-Nielsen, O.E. and Stelzer, R. (2007). Positive-definite matrix processes of finite variation. Probab. Math. Statist. 27 3–43.
  • [6] Bauwens, L., Laurent, S. and Rombouts, J.V.K. (2006). Multivariate GARCH models: A survey. J. Appl. Econometrics 21 79–109.
  • [7] Bhatia, R. (1997). Matrix Analysis. Graduate Texts in Mathematics 169. New York: Springer.
  • [8] Billingsley, P. (1999). Convergence of Probability Measures. New York: Wiley.
  • [9] Blæsild, P. and Jensen, J.L. (1981). Multivariate distributions of hyperbolic type. In Statistical Distributions in Scientific Work – Proceedings of the NATO Advanced Study Institute Held at the Université degli Studi di Trieste, Triest, Italy, July 10–August 1, 1980, Vol. 4 (C. Taillie, G.P. Patil and B.A. Baldessari, eds.) 45–66. Dordrecht: Reidel.
  • [10] Brockwell, P., Chadraa, E. and Lindner, A. (2006). Continuous time GARCH processes. Ann. Appl. Probab. 16 790–826.
  • [11] Bru, M.-F. (1991). Wishart processes. J. Theoret. Probab. 4 725– 751.
  • [12] Da Fonseca, J., Grasselli, M. and Tebaldi, C. (2007). Option pricing when correlations are stochastic: An analytical framework. Review of Derivatives Research 10 151–180.
  • [13] da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes Series 229. Cambridge: Cambridge Univ. Press.
  • [14] Donati-Martin, C., Doumerc, Y., Matsumoto, H. and Yor, M. (2004). Some properties of the Wishart processes and a matrix extension of the Hartman–Watson laws. Publ. Res. Inst. Math. Sci. 40 1385–1412.
  • [15] Dynkin, E.B. (1965). Markov Processes Volume I. Grundlehren der Mathematischen Wissenschaften 121. Berlin: Springer.
  • [16] Engle, R.F. and Kroner, K.F. (1995). Multivariate simultaneous generalized ARCH. Econometric Theory 11 122–150.
  • [17] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
  • [18] Gihman, I.I. and Skorohod, A.V. (1975). The Theory of Stochastic Processes II. Grundlehren der Mathematischen Wissenschaften 218. Berlin: Springer.
  • [19] Gourieroux, C. (2006). Continuous time Wishart process for stochastic risk. Econometric Rev. 25 177–217.
  • [20] Grasselli, M. and Tebaldi, C. (2008). Solvable affine term structure models. Math. Finance 18 135–153.
  • [21] Hafner, C.M. (2003). Fourth moment structure of multivariate GARCH models. Journal of Financial Econometrics 1 26–54.
  • [22] Haug, S., Klüppelberg, C., Lindner, A. and Zapp, M. (2007). Method of moment estimation in the COGARCH(1, 1) model. Econom. J. 10 320–341.
  • [23] Horn, R.A. and Johnson, C.R. (1985). Matrix Analysis. Cambridge: Cambridge Univ. Press.
  • [24] Horn, R.A. and Johnson, C.R. (1991). Topics in Matrix Analysis. Cambridge: Cambridge Univ. Press.
  • [25] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Berlin: Springer.
  • [26] 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.
  • [27] Klüppelberg, C., Lindner, A. and Maller, R. (2006). Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models. In The Shiryaev Festschrift: From Stochastic Calculus to Mathematical Finance (Y. Kabanov, R. Lipster and J. Stoyanov, eds.) 393–419. Berlin: Springer.
  • [28] Kyprianou, A.E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Berlin: Universitext. Springer.
  • [29] Maejima, M. and Rosiński, J. (2002). Type G distributions on ℝd. J. Theoret. Probab. 15 323–341.
  • [30] Magnus, J.R. and Neudecker, H. (1979). The commutation matrix: Some properties and applications. Ann. Statist. 7 381–394.
  • [31] Masuda, H. (2004). On multidimensional Ornstein–Uhlenbeck processes driven by a general Lévy process. Bernoulli 10 97–120.
  • [32] Métivier, M. (1982). Semimartingales: A Course on Stochastic Processes. De Gruyter Studies in Mathematics 2. Berlin: Walter de Gruyter.
  • [33] Métivier, M. and Pellaumail, J. (1980). Stochastic Integration. New York: Academic Press.
  • [34] Pigorsch, C. and Stelzer, R. (2009a). A multivariate Ornstein–Uhlenbeck type stochastic volatility model. Submitted. Available at
  • [35] Pigorsch, C. and Stelzer, R. (2009b). On the definition, stationary distribution and second order structure of positive semi-definite Ornstein–Uhlenbeck type processes. Bernoulli. To appear.
  • [36] Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. New York: Springer.
  • [37] Reiss, M., Riedle, M. and van Gaans, O. (2006). Delay differential equations driven by Lévy processes: Stationarity and Feller properties. Stochastic Process. Appl. 116 1409–1432.
  • [38] Reiss, M., Riedle, M. and van Gaans, O. (2007). On Émery’s inequality and a variation-of-constants formula. Stoch. Anal. Appl. 25 353–379.
  • [39] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
  • [40] Stelzer, R.J. (2007). Multivariate continuous time stochastic volatility models driven by a Lévy process. Ph.D. thesis, Fakultät für Mathematik, Technische Universität München, Garching, Germany. Available at