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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 375480, 9 pages
The Local Time of the Fractional Ornstein-Uhlenbeck Process
Department of Mathematics, Anhui Normal University, 1 East Beijing Road, Wuhu 241000, China
Received 11 June 2013; Revised 11 August 2013; Accepted 11 August 2013
Academic Editor: Mark McKibben
Copyright © 2013 Guangjun Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, New York, NY, USA, 1999.
- J.-F. Gouyet, Physics and Fractal Structures, Springer, Berlin, Germany, 1996.
- D. Nualart, The Malliavin Calculus and Related Topics, Springer, New York, NY, USA, 2006.
- F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2008.
- Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2008.
- W. Willinger, M. S. Taqqu, and V. Teverovsky, “Stock market prices and long-range dependence,” Finance Stochastics, vol. 3, no. 1, pp. 1–13, 1999.
- B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968.
- P. Cheridito, H. Kawaguchi, and M. Maejima, “Fractional Ornstein-Uhlenbeck processes,” Electronic Journal of Probability, vol. 8, article 3, pp. 1–14, 2003.
- S. C. Lim and S. V. Muniandy, “Generalized Ornstein-Uhlenbeck processes and associated self-similar processes,” Journal of Physics A, vol. 36, no. 14, pp. 3961–3982, 2003.
- R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004.
- L. Yan, Y. Lu, and Z. Xu, “Some properties of the fractional Ornstein-Uhlenbeck process,” Journal of Physics A, vol. 41, no. 14, Article ID 145007, 17 pages, 2008.
- L. Yan and M. Tian, “On the local times of fractional Ornstein-Uhlenbeck process,” Letters in Mathematical Physics, vol. 73, no. 3, pp. 209–220, 2005.
- S. R. S. Varadhan, “Appendix to euclidean quantum field theory by K. Symanzk,” in Local Quantum Theory, R. Jost, Ed., Academic Press, New York, NY, USA, 1969.
- R. J. Adler, The Geometry of Random Fields, John Wiley & Sons, New York, NY, USA, 1981.
- D. Geman and J. Horowitz, “Occupation densities,” The Annals of Probability, vol. 8, no. 1, pp. 1–67, 1980.
- Y. Xiao, “Local times and related properties of multidimensional iterated Brownian motion,” Journal of Theoretical Probability, vol. 11, no. 2, pp. 383–408, 1998.
- Y. Xiao, “Properties of local-nondeterminism of Gaussian and stable random fields and their applications,” Annales de la Faculté des Sciences de Toulouse Série 6, vol. 15, no. 1, pp. 157–193, 2006.
- Y. Xiao, “Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields,” Probability Theory and Related Fields, vol. 109, no. 1, pp. 129–157, 1997.
- Y. Hu, “Hausdorff and packing measures of the level sets of iterated Brownian motion,” Journal of Theoretical Probability, vol. 12, no. 2, pp. 313–346, 1999.
- B. Boufoussi, M. Dozzi, and R. Guerbaz, “On the local time of multifractional Brownian motion,” Stochastics, vol. 78, no. 1, pp. 33–49, 2006.
- B. Boufoussi, M. Dozzi, and R. Guerbaz, “Sample path properties of the local time of multifractional Brownian motion,” Bernoulli, vol. 13, no. 3, pp. 849–867, 2007.
- S. M. Berman, “Local nondeterminism and local times of Gaussian processes,” Indiana University Mathematics Journal, vol. 23, pp. 69–94, 1974.
- J. P. Nolan, “Local nondeterminism and local times for stable processes,” Probability Theory and Related Fields, vol. 82, no. 3, pp. 387–410, 1989.
- Y. Hu, “Self-intersection local time of fractional Brownian motions—via chaos expansion,” Journal of Mathematics of Kyoto University, vol. 41, no. 2, pp. 233–250, 2001.
- S. Watanabe, Stochastic Differential Equation and Malliavin Calculus, Tata Institute of Fundamental Reaearch, Springer, New York, NY, USA, 1984.
- S. M. Berman, “Local times and sample function properties of stationary Gaussian processes,” Transactions of the American Mathematical Society, vol. 137, pp. 277–299, 1969.
- Y. Jiang and Y. Wang, “On the collision local time of fractional Brownian motions,” Chinese Annals of Mathematics B, vol. 28, no. 3, pp. 311–320, 2007.
- L. An and L. Yan, “Smoothness for the collision local time of fractional Brownian motion,” Preprint.