Since the pioneering work of Hurst, and Mandelbrot, the fractional brownian motions have played and increasingly important role in many fields of application such as hydrology, economics and telecommunications. For every value of the Hurst index H ∈ (0,1) we define a stochastic integral with respect to fractional Brownian motion of index H. This process is called a (standard) fractional Brownian motion with Hurst parameter H. To simplify the presentation, it is always assumed that the fractional Brownian motion is 0 at t=0. If H = 1/2, then the corresponding fractional Brownian motion is the usual standard Brownian motion. If 1/2 < H < 1, Fractional Brownian motion (FBM) is neither a finite variation nor a semi-martingale. Consequently, the standard Ito calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2 < H < 1. The classic methods (Itô and Stiliege) are excluted. The most studied case is that where H is between 0 and 1/2. Several attempts to define the stochastic integral are made. But so far some difficulties subjust. We give in this paper, several construction methods. So for the construction, we will use other tools to deal with such situations.
| Published in | American Journal of Applied Mathematics (Volume 9, Issue 5) |
| DOI | 10.11648/j.ajam.20210905.11 |
| Page(s) | 156-164 |
| 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), 2024. Published by Science Publishing Group |
Wiener Integral, Fractional Brownian Motion, Martingale, Processus d’Ito
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APA Style
Diop Bou, Ba Demba Bocar, Thioune Moussa. (2021). Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion. American Journal of Applied Mathematics, 9(5), 156-164. https://doi.org/10.11648/j.ajam.20210905.11
ACS Style
Diop Bou; Ba Demba Bocar; Thioune Moussa. Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion. Am. J. Appl. Math. 2021, 9(5), 156-164. doi: 10.11648/j.ajam.20210905.11
AMA Style
Diop Bou, Ba Demba Bocar, Thioune Moussa. Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion. Am J Appl Math. 2021;9(5):156-164. doi: 10.11648/j.ajam.20210905.11
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Download@article{10.11648/j.ajam.20210905.11,
author = {Diop Bou and Ba Demba Bocar and Thioune Moussa},
title = {Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion},
journal = {American Journal of Applied Mathematics},
volume = {9},
number = {5},
pages = {156-164},
doi = {10.11648/j.ajam.20210905.11},
url = {https://doi.org/10.11648/j.ajam.20210905.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210905.11},
abstract = {Since the pioneering work of Hurst, and Mandelbrot, the fractional brownian motions have played and increasingly important role in many fields of application such as hydrology, economics and telecommunications. For every value of the Hurst index H ∈ (0,1) we define a stochastic integral with respect to fractional Brownian motion of index H. This process is called a (standard) fractional Brownian motion with Hurst parameter H. To simplify the presentation, it is always assumed that the fractional Brownian motion is 0 at t=0. If H = 1/2, then the corresponding fractional Brownian motion is the usual standard Brownian motion. If 1/2 H H ô and Stiliege) are excluted. The most studied case is that where H is between 0 and 1/2. Several attempts to define the stochastic integral are made. But so far some difficulties subjust. We give in this paper, several construction methods. So for the construction, we will use other tools to deal with such situations.},
year = {2021}
}
TY - JOUR T1 - Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion AU - Diop Bou AU - Ba Demba Bocar AU - Thioune Moussa Y1 - 2021/09/03 PY - 2021 N1 - https://doi.org/10.11648/j.ajam.20210905.11 DO - 10.11648/j.ajam.20210905.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 156 EP - 164 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20210905.11 AB - Since the pioneering work of Hurst, and Mandelbrot, the fractional brownian motions have played and increasingly important role in many fields of application such as hydrology, economics and telecommunications. For every value of the Hurst index H ∈ (0,1) we define a stochastic integral with respect to fractional Brownian motion of index H. This process is called a (standard) fractional Brownian motion with Hurst parameter H. To simplify the presentation, it is always assumed that the fractional Brownian motion is 0 at t=0. If H = 1/2, then the corresponding fractional Brownian motion is the usual standard Brownian motion. If 1/2 H H ô and Stiliege) are excluted. The most studied case is that where H is between 0 and 1/2. Several attempts to define the stochastic integral are made. But so far some difficulties subjust. We give in this paper, several construction methods. So for the construction, we will use other tools to deal with such situations. VL - 9 IS - 5 ER -
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