Electronic Journal of Probability

On the Exponentials of Fractional Ornstein-Uhlenbeck Processes

Muneya Matsui and Narn-Rueih Shieh

Full-text: Open access

Abstract

We study the correlation decay and the expected maximal increment (Burkholder-Davis-Gundy type inequalities) of the exponential process determined by a fractional Ornstein-Uhlenbeck process. The method is to apply integration by parts formula on integral representations of fractional Ornstein-Uhlenbeck processes, and also to use Slepian's inequality. As an application, we attempt Kahane's T-martingale theory based on our exponential process which is shown to be of long memory.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 23, 594-611.

Dates
Accepted: 27 February 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819484

Digital Object Identifier
doi:10.1214/EJP.v14-628

Mathematical Reviews number (MathSciNet)
MR2486815

Zentralblatt MATH identifier
1191.60048

Subjects
Primary: 60G17: Sample path properties 60G15: Gaussian processes
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 60G10: Stationary processes

Keywords
Long memory (Long range dependence) Fractional Brownian motion Fractional Ornstein-Uhlenbeck process Exponential process Burkholder-Davis-Gundy inequalities

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Matsui, Muneya; Shieh, Narn-Rueih. On the Exponentials of Fractional Ornstein-Uhlenbeck Processes. Electron. J. Probab. 14 (2009), paper no. 23, 594--611. doi:10.1214/EJP.v14-628. https://projecteuclid.org/euclid.ejp/1464819484


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References

  • V.V. Anh, N.N. Leonenko and N.-R. Shieh. Multifractal products of stationary diffusion processes. To appear in Stochastic Anal. Appl.
  • Ph. Carmona, F. Petit and M. Yor. On the distribution and asymptotic results for exponential functionals of Levy processes. In Exponential Functionals and Principal Values Related to Brownian Motion (ed. M. Yor), 1997, Biblioteca de la Revista Matematica Iberoamericana, pp. 73-126.
  • Ph. Carmona, F. Petit and M. Yor. Exponential functionals of Levy processes.@ In Levy Processes: Theory and Applications, eds. O.E. Barndorff-Nielsen, T. Mikosch and S.I. Resnick, Birkhauser, 2001, pp. 41-55.
  • P. Cheridito, H. Kawaguchi and M. Maejima. Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8 (2003), paper 3, 1-14.
  • P. Embrechts and M. Maejima. Selfsimilar Processes. Princeton Series in Applied Mathematics, Princeton University Press, 2002.
  • I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Translated from the Russian. Sixth edition. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. Academic Press, Inc., San Diego, CA, 2000.
  • S.E. Graversen and G. Peskir. Maximal inequalities for the Ornstein-Uhlenbeck process. Proc. Amer. Math. Soc. 128 (2000), 3035-3041.
  • J.P. Kahane. Random coverings and multiplicative processes. Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998) Progr. Probab. 46, Birkhauser, Basel, 2000. pp.125-146
  • M.R. Leadbetter, G. Lindgren and H. Rootzen. Extremes and Related Properties of Random Sequences and processes. Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1983.
  • P. Mannersalo, I. Norros and R.H. Riedi. Multifractal products of stochastic processes: construction and some basic properties. Adv. in Appl. Probab. 34 (2002), 888-903.
  • J. Memin, Y. Mishura and E. Valkeila. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51 (2001), 197-206.
  • Y.S. Mishura. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008.
  • A. Novikov and E. Valkeila. On some maximal inequalities for fractional Brownian motions. Statist. Probab. Lett. 44 (1999), 47-54.
  • J. Peyriere. Recent results on Mandelbrot multiplicative cascades. Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998) Progr. Probab. 46 Birkhauser, Basel, 2000, pp.147-159.
  • Pipiras, V. and M.S. Taqqu. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000), 251-291.
  • D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1999,
  • A.A. Ruzmaikina. Stieltjes integrals of Holder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 (2000), 1049-1069.
  • G. Samorodnitsky. Long range dependence. Found. Trends Stoch. Syst. 1 (2006), 163-257.
  • G. Samorodnitsky and M.S. Taqqu. Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance (1994), Chapman and Hall, New York.
  • P.E. Protter. Stochastic Integration and Differential Equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004.
  • D. Slepian. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962), 463-501.
  • R.L. Wheeden and A. Zygmund. Measure and Integral, Marcel Dekker, New York-Basel, 1977.