On the Exponentials of Fractional Ornstein-Uhlenbeck Processes

Muneya Matsui (Department of Mathematics, Keio University)
Narn-Rueih Shieh (Department of Mathematics, National Taiwan University)

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.

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Pages: 594-611

Publication Date: February 27, 2009

DOI: 10.1214/EJP.v14-628

References

  1. V.V. Anh, N.N. Leonenko and N.-R. Shieh. Multifractal products of stationary diffusion processes. To appear in Stochastic Anal. Appl. Math. Review number not available.
  2. 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. MR1648657
  3. 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. MR1833691
  4. P. Cheridito, H. Kawaguchi and M. Maejima. Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8 (2003), paper 3, 1-14. MR1961165
  5. P. Embrechts and M. Maejima. Selfsimilar Processes. Princeton Series in Applied Mathematics, Princeton University Press, 2002. MR1920153
  6. 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. MR1773820
  7. S.E. Graversen and G. Peskir. Maximal inequalities for the Ornstein-Uhlenbeck process. Proc. Amer. Math. Soc. 128 (2000), 3035-3041. MR1664394
  8. 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 MR1785624
  9. 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. MR0691492
  10. 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. MR1938947
  11. 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. MR1822771
  12. Y.S. Mishura. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, 1929. Springer-Verlag, Berlin, 2008. MR2378138
  13. A. Novikov and E. Valkeila. On some maximal inequalities for fractional Brownian motions. Statist. Probab. Lett. 44 (1999), 47-54. MR1706311
  14. 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. MR1785625
  15. Pipiras, V. and M.S. Taqqu. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000), 251-291. MR1790083
  16. 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, MR1725357
  17. A.A. Ruzmaikina. Stieltjes integrals of Holder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 (2000), 1049-1069. MR1798553
  18. G. Samorodnitsky. Long range dependence. Found. Trends Stoch. Syst. 1 (2006), 163-257. MR2379935
  19. G. Samorodnitsky and M.S. Taqqu. Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance (1994), Chapman and Hall, New York. MR1280932
  20. 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. MR2020294
  21. D. Slepian. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962), 463-501. MR0133183
  22. R.L. Wheeden and A. Zygmund. Measure and Integral, Marcel Dekker, New York-Basel, 1977. MR0492146
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