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Tytuł artykułu

Occupation time problem for multifractional Brownian motion

Wybrane pełne teksty z tego czasopisma
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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions.
Rocznik
Strony
99--113
Opis fizyczny
Bibliogr. 21 poz.
Twórcy
  • Laboratoire de Modélisation, Stochastique et Déterministe et URAC (04), Faculté des Sciences Oujda, Maroc
autor
  • Laboratoire MAEGE and Department SMAEG, FSJES Ain Sebaa, Hassan II University of Casablanca, Morocco
  • Department of Statistics, College of Business and Economics, United Arab Emirates University
autor
  • Laboratoire de Modélisation, Stochastique et Déterministe, Département de Mathématiques, Faculté des Sciences Oujda, Université Mohammed Premier, Maroc
Bibliografia
  • [1] M. Ait Ouahra, Weak convergence to fractional Brownian motion in some anisotropic Besov space, Ann. Math. Blaise Pascal 11 (1) (2004), pp. 1-17.
  • [2] M. Ait Ouahra and M. Eddahbi, Théorèmes limites pour certaines fonctionnelles associées aux processus stables sur l’espace de Hölder, Publ. Mat. 45 (2) (2001), pp. 371-386.
  • [3] M. Ait Ouahra and M. Ouali, Occupation time problems for fractional Brownian motion and some other self-similar processes, Random Oper. Stoch. Equ. 17 (1) (2009), pp. 69-89.
  • [4] A. Benassi, S. Jaffard, and D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoam. 13 (1) (1997), pp. 19-90.
  • [5] S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 (1973/74), pp. 69-94.
  • [6] B. Boufoussi, M. Dozzi, and R. Guerbaz, On the local time of multifractional Brownian motion, Stochastics 78 (1) (2006), pp. 33-49.
  • [7] B. Boufoussi, M. Dozzi, and R. Guerbaz, Sample path properties of the local time of multifractional Brownian motion, Bernoulli 13 (3) (2007), pp. 849-867.
  • [8] M. Eddahbi and J. Vives, Chaotic expansion and smoothness of some functionals of the fractional Brownian motion, J. Math. Kyoto Univ. 43 (2) (2003), pp. 349-368.
  • [9] H. Ezawa, J. R. Klauder, and L. A. Shepp, Vestigial effects of singular potentials in diffusion theory and quantum mechanics, J. Math. Phys. 16 (1975), pp. 783-799.
  • [10] P. J. Fitzsimmons and R. K. Getoor, Limit theorems and variation properties for fractional derivatives of the local time of a stable process, Ann. Inst. H. Poincaré Probab. Statist. 28 (2) (1992), pp. 311-333.
  • [11] M. Fukushima, A decomposition of additive functionals of finite energy, Nagoya Math. J. 74 (1979), pp. 137-168.
  • [12] I. I. Gīhman and A. V. Skorohod, The Theory of Stochastic Processes. I, Springer, New York-Heidelberg 1974.
  • [13] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1) (1928), pp. 565-606.
  • [14] M. Jolis and N. Viles, Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theoret. Probab. 20 (2) (2007), pp. 133-152.
  • [15] J. Lévy-Véhel, Introduction to the multifractal analysis of images, in: Fractal Image Encoding and Analysis, Y. Fisher (Ed.), Springer, New York 1995.
  • [16] J. Lévy-Véhel and R. F. Peltier, Multifractional Brownian motion: Definition and preliminary results, [Research Report], RR-2645, INRIA, 1995.
  • [17] M. Meerschaert, D. Wu, and Y. Xiao, Local times of multifractional Brownian sheets, Bernoulli 14 (3) (2008), pp. 865-898.
  • [18] B. Pesquet-Popescu and J. Lévy-Véhel, Stochastic fractal models for image processing, IEEE Signal Processing Magazine 19 (5) (2002), pp. 48-62.
  • [19] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon 1993.
  • [20] N. Shieh, Limit theorems for local times of fractional Brownian motions and some other self-similar processes, J. Math. Kyoto Univ. 36 (4) (1996), pp. 641-652.
  • [21] T. Yamada, On some limit theorems for occupation times of one-dimensional Brownian motion and its continuous additive functionals locally of zero energy, J. Math. Kyoto Univ. 26 (2) (1986), pp. 309-322.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-cf317dc6-a773-4118-b002-47a9cdae2288
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