• Bernoulli
  • Volume 17, Number 3 (2011), 916-941.

Quasi Ornstein–Uhlenbeck processes

Ole E. Barndorff-Nielsen and Andreas Basse-O’Connor

Full-text: Open access


The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of the Wold–Karhunen decomposition theorem, such solutions are, in principle, representable as a moving average (plus a drift-like term) but the kernel in the moving average is generally not available in explicit form. A class of cases is determined where an explicit expression of the kernel can be given, and this is used to obtain information on the asymptotic behavior of the associated autocorrelation functions, both for small and large lags. Applications to Gaussian- and Lévy-driven fractional Ornstein–Uhlenbeck processes are presented. A Fubini theorem for Lévy bases is established as an element in the derivations.

Article information

Bernoulli Volume 17, Number 3 (2011), 916-941.

First available in Project Euclid: 7 July 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

fractional Ornstein–Uhlenbeck processes Fubini theorem for Lévy bases Langevin equations stationary processes


Barndorff-Nielsen, Ole E.; Basse-O’Connor, Andreas. Quasi Ornstein–Uhlenbeck processes. Bernoulli 17 (2011), no. 3, 916--941. doi:10.3150/10-BEJ311.

Export citation


  • [1] Barndorff-Nielsen, O.E. (2001). Superposition of Ornstein–Uhlenbeck type processes. Theory Probab. Appl. 46 175–194.
  • [2] Barndorff-Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2011). Multipower variation for Brownian semistationary processes. Bernoulli. To appear. Available at
  • [3] Barndorff-Nielsen, O.E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241.
  • [4] Barndorff-Nielsen, O.E. and Shephard, N. (2011). Financial Volatility in Continuous Time. Cambridge: Cambridge Univ. Press. To appear.
  • [5] Barndorff-Nielsen, O.E. and Shiryaev, A.N. (2010). Change of Time and Change of Measure. Singapore: World Scientific. To appear.
  • [6] Barndorff-Nielsen, O.E. and Stelzer, R. (2011). Multivariate supOU processes. Ann. Appl. Probab. 21 140–182.
  • [7] Basse, A. (2008). Gaussian moving averages and semimartingales. Electron. J. Probab. 13 1140–1165.
  • [8] Basse, A. (2009). Spectral representation of Gaussian semimartingales. J. Theoret. Probab. 22 811–826.
  • [9] Basse, A. and Pedersen, J. (2009). Lévy driven moving averages and semimartingales. Stochastic Process. Appl. 119 2970–2991.
  • [10] Basse-O’Connor, A. (2010). Representation of Gaussian semimartingales with applications to the covariance function. Stochastics 82 381–401.
  • [11] Biagini, F., Hu, Y., Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. London: Springer.
  • [12] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [13] Carr, P., Geman, H., Madan, D.B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13 345–382.
  • [14] Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein–Uhlenbeck processes. Electron. J. Probab. 8 1–14 (electronic).
  • [15] Cohn, D.L. (1972). Measurable choice of limit points and the existence of separable and measurable processes. Z. Wahrsch. Verw. Gebiete 22 161–165.
  • [16] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Boca Raton, FL: Chapman & Hall/CRC.
  • [17] Doob, J.L. (1990). Stochastic Processes. New York: Wiley.
  • [18] Doukhan, P., Oppenheim, G. and Taqqu, M.S., eds. (2003). Theory and Applications of Long-Range Dependence. Boston, MA: Birkhäuser.
  • [19] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library 24. Amsterdam: North-Holland.
  • [20] Karhunen, K. (1950). Über die Struktur stationärer zufälliger Funktionen. Ark. Mat. 1 141–160.
  • [21] Khamsi, M.A. (1996). A convexity property in modular function spaces. Math. Japon. 44 269–279.
  • [22] Knight, F.B. (1992). Foundations of the Prediction Process. Oxford Studies in Probability 1. New York: Oxford Univ. Press.
  • [23] Maejima, M. and Yamamoto, K. (2003). Long-memory stable Ornstein–Uhlenbeck processes. Electron. J. Probab. 8 1–18 (electronic).
  • [24] Marcus, M.B. and Rosiński, J. (2001). L1-norms of infinitely divisible random vectors and certain stochastic integrals. Electron. Commun. Probab. 6 15–29 (electronic).
  • [25] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12 1099–1126.
  • [26] Musielak, J. (1983). Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034. Berlin: Springer.
  • [27] Nualart, D. (2006). Fractional Brownian motion: Stochastic calculus and applications. In International Congress of Mathematicians III 1541–1562. Zürich: Eur. Math. Soc.
  • [28] Pérez-Abreu, V. and Rocha-Arteaga, A. (1997). On stable processes of bounded variation. Statist. Probab. Lett. 33 69–77.
  • [29] Protter, P.E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Berlin: Springer.
  • [30] Rajput, B.S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Related Fields 82 451–487.
  • [31] Rogers, L.C.G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 1. Foundations. Cambridge: Cambridge Univ. Press.
  • [32] Rolewicz, S. (1985). Metric Linear Spaces, 2nd ed. Mathematics and Its Applications (East European Series) 20. Dordrecht: D. Reidel Publishing Co.
  • [33] Rosiński, J. (1986). On stochastic integral representation of stable processes with sample paths in Banach spaces. J. Multivariate Anal. 20 277–302.
  • [34] Rosiński, J. (2007). Spectral representation of infinitely divisible processes and injectivity of the ϒ-transformation. In 5th International Conference on Lévy Processes: Theory and Applications, Copenhagen 2007. Available at
  • [35] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman & Hall.
  • [36] Sato, K. and Yamazato, M. (1983). Stationary processes of Ornstein–Uhlenbeck type. In Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Math. 1021 541–551. Berlin: Springer.
  • [37] Shao, Y. (1995). The fractional Ornstein–Uhlenbeck process as a representation of homogeneous Eulerian velocity turbulence. Phys. D 83 461–477.
  • [38] Surgailis, D., Rosiński, J., Mandekar, V. and Cambanis, S. (1998). On the mixing structure of stationary increments and self-similar SαS processes. Preprint.
  • [39] Wolfe, S.J. (1982). On a continuous analogue of the stochastic difference equation Xn = ρXn−1 + Bn. Stochastic Process. Appl. 12 301–312.