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Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion

Received: 17 June 2020     Accepted: 16 October 2020     Published: 4 December 2020
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Abstract

In this paper we define the multifractional Brownian motion and we give some properties. we study the uniform Convergence of the Serie expansion. After having determined the covariance function, we give in proposition 2 another proof of almost sure uniform convergence on compact K of the series. We will finish by showing that the m.B.f is locally astymptotically self-similar, with field or fractional Brownian field with Hurst exposant H. One of the problem, for application of multifractional Brownian motion, is the regularity of the function. In the filtered white noise model the increments are no more homogeneous as in fractional Brownian field case. It is obvious when we consider the tangent field associated with a function. Still the multifractional function in the previous model is constant and it is not convient for many applications. We show the uniform convergence of the series on K. We deduce from the previous questions the almost sure uniform convergence of the series to a mBm.

Published in Applied and Computational Mathematics (Volume 9, Issue 6)
DOI 10.11648/j.acm.20200906.14
Page(s) 195-200
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Multifractional Brownian Motion, Uniform Convergence, Series Expansion

References
[1] A. Ayache, A. Benassi, S. Cohen, J. Levy- Vehel: Regularity and Identification of Generalized Multifractional Gaussian Processes, in: Lecture Notes in Math, vol 1857 Spinger, Berlin, 2005, pp 290-312.
[2] A. Ayache, S. Cohen, J. L. Vehel the covariance structure of multifractional Brownian motion in: IEEE ICASSP 2000.
[3] A. Begyn, Asymptotic expansion and central limit theorem for quadratic variation of Gaussian processes, stochastic Process. Appl 117 (2007).
[4] A. Brouste J. Ista and S. Lambert- Lacroix On fractional Gaussian random fields simulation J. Stat. Soft 2007.
[5] A. Stoev Stilian, Murad S. Taqqu: How rich in the class multifractional brownian motion, stochastic Processes and their applications 116, 2006, 200-221.
[6] A. Benassi, S Jaffard, D. Roux, Elliptic Gaussian fandom processes, rev. Math Iber 13 (1) 1997 19-90.
[7] Ba Demba Bocar: On the fractional Brownien motion: Hausdorf dimension and Fourier expansion international journal of advances in applied mathematical and mechanics vol 5 pp 53-59 (2017).
[8] Ba Demba Bocar: Fractional operators and Applications to fractional martingal international journal of advances in applied mathematical and mechanics vol 5 (2018).
[9] C. Lacaux Real harmonizable multifractional Levy motions. Ann. Inst. Poincarre Proba-Stat 2004
[10] C. Locaux Fields with exceptional tangent fields. J. theorie Proba 2005.
[11] Mandelbrot and J. W. Van Ness, Fractional Brownian motion, fractional noises and applications STAN Rev, 10 422-437, 1968.
[12] M. Clausel. Lacunary fractional Brownian motion E SAIM. 2012.
[13] P. K. Frizand N. B. Victor, Multidimensional Stochastic Processes as Rough Paths: theory and Applications. Cambridge University Press, 2010.
[14] R. F Peltier, J. L. Vehel, Multifractional Brownian motion, definition and preliminary results, technical Report 2645, INRIA, le chesnay France, 1995.
[15] Serge Cohen, Jacques Istas, Fractional fields and applications. Springer 2010.
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  • APA Style

    BA Demba Bocar. (2020). Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion. Applied and Computational Mathematics, 9(6), 195-200. https://doi.org/10.11648/j.acm.20200906.14

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    ACS Style

    BA Demba Bocar. Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion. Appl. Comput. Math. 2020, 9(6), 195-200. doi: 10.11648/j.acm.20200906.14

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    AMA Style

    BA Demba Bocar. Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion. Appl Comput Math. 2020;9(6):195-200. doi: 10.11648/j.acm.20200906.14

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  • @article{10.11648/j.acm.20200906.14,
      author = {BA Demba Bocar},
      title = {Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion},
      journal = {Applied and Computational Mathematics},
      volume = {9},
      number = {6},
      pages = {195-200},
      doi = {10.11648/j.acm.20200906.14},
      url = {https://doi.org/10.11648/j.acm.20200906.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200906.14},
      abstract = {In this paper we define the multifractional Brownian motion and we give some properties. we study the uniform Convergence of the Serie expansion. After having determined the covariance function, we give in proposition 2 another proof of almost sure uniform convergence on compact K of the series. We will finish by showing that the m.B.f is locally astymptotically self-similar, with field or fractional Brownian field with Hurst exposant H. One of the problem, for application of multifractional Brownian motion, is the regularity of the function. In the filtered white noise model the increments are no more homogeneous as in fractional Brownian field case. It is obvious when we consider the tangent field associated with a function. Still the multifractional function in the previous model is constant and it is not convient for many applications. We show the uniform convergence of the series on K. We deduce from the previous questions the almost sure uniform convergence of the series to a mBm.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion
    AU  - BA Demba Bocar
    Y1  - 2020/12/04
    PY  - 2020
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    T2  - Applied and Computational Mathematics
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    PB  - Science Publishing Group
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    AB  - In this paper we define the multifractional Brownian motion and we give some properties. we study the uniform Convergence of the Serie expansion. After having determined the covariance function, we give in proposition 2 another proof of almost sure uniform convergence on compact K of the series. We will finish by showing that the m.B.f is locally astymptotically self-similar, with field or fractional Brownian field with Hurst exposant H. One of the problem, for application of multifractional Brownian motion, is the regularity of the function. In the filtered white noise model the increments are no more homogeneous as in fractional Brownian field case. It is obvious when we consider the tangent field associated with a function. Still the multifractional function in the previous model is constant and it is not convient for many applications. We show the uniform convergence of the series on K. We deduce from the previous questions the almost sure uniform convergence of the series to a mBm.
    VL  - 9
    IS  - 6
    ER  - 

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  • UFR SET Thies of University Senegal, Thies, Senegal

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