Fractional Ornstein-Uhlenbeck processes

Patrick Cheridito (ETH Zurich)
Hideyuki Kawaguchi (Keio University and Sumitomo Mitsui Banking Corporation)
Makoto Maejima (Keio University)

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.

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Pages: 1-14

Publication Date: February 15, 2003

DOI: 10.1214/EJP.v8-125

References

  1. Doob, J.L. (1942), The Brownian movement and stochastic equations, Ann. of Math. (2) 43, 351-369. Math. Review 4,17d
  2. Embrechts, P. and Maejima, M. (2002), Selfsimilar Processes, Princeton Series in Applied Mathematics, Princeton University Press. Math. Review 1 920 153
  3. Karatzas, I. and Shreve, S.E. (1991), Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, Springer-Verlag, New York. Math. Review 92h:60127
  4. Lamperti, J.W. (1962), Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104, 62-78. Math. Review 25 #1575
  5. Langevin, P. (1908), Sur la théorie du mouvement brownien, C.R. Acad. Sci. Paris 146, 530-533.
  6. Pipiras, V. and Taqqu, M. (2000), Integration questions related to fractional Brownian motion, Prob. Th. Rel. Fields 118, 121-291. Math. Review 2002c:60091
  7. Protter, P. (1990), Stochastic Integration and Differential Equations, Springer-Verlag, Berlin. Math. Review 91i:60148
  8. Samorodnitsky, G. and Taqqu, M.S. (1994), Stable Non-Gaussian Random Processes, Chapman & Hall, New York. Math. Review 95f:60024
  9. Uhlenbeck, G.E. and Ornstein, L.S. (1930), On the theory of the Brownian motion, Physical Review 36, 823-841.
  10. Wheeden, R.L. and Zygmund, A. (1977), Measure and Integral, Marcel Dekker, New York-Basel. Math. Review 58 #11295
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