Electronic Journal of Probability

Fractional Ornstein-Uhlenbeck processes

Patrick Cheridito, Hideyuki Kawaguchi, and Makoto Maejima

Full-text: Open access

Abstract

The classical stationary Ornstein-Uhlenbeck process can be obtained in two different ways. On the one hand, it is a stationary solution of the Langevin equation with Brownian motion noise. On the other hand, it can be obtained from Brownian motion by the so called Lamperti transformation. We show that the Langevin equation with fractional Brownian motion noise also has a stationary solution and that the decay of its auto-covariance function is like that of a power function. Contrary to that, the stationary process obtained from fractional Brownian motion by the Lamperti transformation has an auto-covariance function that decays exponentially.

Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 3, 14 p.

Dates
First available in Project Euclid: 23 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464037576

Digital Object Identifier
doi:10.1214/EJP.v8-125

Mathematical Reviews number (MathSciNet)
MR1961165

Zentralblatt MATH identifier
1065.60033

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes 45F05: Systems of nonsingular linear integral equations

Keywords
Fractional Brownian motion Langevin equation Long-range dependence Selfsimilar processes Lampertitransformation

Citation

Cheridito, Patrick; Kawaguchi, Hideyuki; Maejima, Makoto. Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8 (2003), paper no. 3, 14 p. doi:10.1214/EJP.v8-125. https://projecteuclid.org/euclid.ejp/1464037576


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References

  • Doob, J.L. (1942), The Brownian movement and stochastic equations, Ann. of Math. (2) 43, 351-369.
  • Embrechts, P. and Maejima, M. (2002), Selfsimilar Processes, Princeton Series in Applied Mathematics, Princeton University Press.
  • Karatzas, I. and Shreve, S.E. (1991), Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, Springer-Verlag, New York.
  • Lamperti, J.W. (1962), Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104, 62-78.
  • Langevin, P. (1908), Sur la théorie du mouvement brownien, C.R. Acad. Sci. Paris 146, 530-533.
  • Pipiras, V. and Taqqu, M. (2000), Integration questions related to fractional Brownian motion, Prob. Th. Rel. Fields 118, 121-291.
  • Protter, P. (1990), Stochastic Integration and Differential Equations, Springer-Verlag, Berlin.
  • Samorodnitsky, G. and Taqqu, M.S. (1994), Stable Non-Gaussian Random Processes, Chapman & Hall, New York.
  • Uhlenbeck, G.E. and Ornstein, L.S. (1930), On the theory of the Brownian motion, Physical Review 36, 823-841.
  • Wheeden, R.L. and Zygmund, A. (1977), Measure and Integral, Marcel Dekker, New York-Basel.