Electronic Journal of Probability

Fractional Ornstein-Uhlenbeck processes

Patrick Cheridito, Hideyuki Kawaguchi, and Makoto Maejima

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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

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

First available in Project Euclid: 23 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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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