The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods

Hamid EBeye; Moussa Thioune; Demba Bocar Ba

doi:doi:10.11648/j.ijtam.20261201.14

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The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods

Hamid EBeye, Moussa Thioune, Demba Bocar Ba^*

Published in International Journal of Theoretical and Applied Mathematics (Volume 12, Issue 1)

Received: 30 November 2025 Accepted: 5 January 2026 Published: 4 February 2026

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Abstract

To overcome some of the limitations of traditional fractional Brownian motion (fBm), multifractional Brownian motion (mBm) was created. The Holder exponent of mBm can vary along the trajectory, unlike fBm, which is advantageous for modeling processes whose regularity varies over time, like internet traffic or photos. This is the main difference between the two processes. Many continuous observations can be modeled by stochastic differential equations governed by mBm, especially in biology and finance. Using a non-stationary multifractional Brownian motion with a time-dependent Hurst parameter as the noise source and a simply linear drift coefficient, we first show the existence and uniqueness of a solution to stochastic differential equations in this article. Next, we examine multifractional Brownian motion (mBm) driven stochastic differential equation models that allow for the simulation of some discontinuity-containing phenomena.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 12, Issue 1)
DOI	10.11648/j.ijtam.20261201.14
Page(s)	31-36
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), 2026. Published by Science Publishing Group

Keywords

Multifractional Brownian Motion, Girsanov’s Theorem, Stochastic Differential Brownian, Riemann-Liouville Multifractional Integral, Multifractional Derivative

References

[1]	A. Benassi, S. Cohen, and J. Istas. “Identifying the multifractional function of a Gaussian process”. Statistics and Probability Letters, 39(4) : 337-345, 1998. 12.
[2]	BA Demba Bocar. and THIOUNE Moussa: Consistency of the drift parameters estimates in the Fractional Brownian Diffusion model and estimation of the hurst parameter by maximum likelihood method. Journal of Computer Science and Applied Mathematics. Vol. 4, No. 1, (2022), pp. 1-14. https://doi.org/10.37418/jcsam.4.1.1
[3]	Ba Demba Bocar, Diop Bou and Thioune Moussa,(2023). An approach to pathwise stochastic integration in fractional Besov-types spaces and by Krylov inequality, Universal Journal of Mathematics and Mathematical Sciences 18(1) (2023), 67-83. http://dx.doi.org/10.17654/2277141723005
[4]	C. Lorenzo and T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn. 29 (2002), pp. 57-98.
[5]	Diop Bou, Ba Demba Bocar, THIOUNE Moussa. Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion. American Journal of Applied Mathematics. Vol. 9, No. 5, (2021), pp. 156-164. https://doi.org/10.11648/j.ajam.20210905.11
[6]	D. Nualart and Y. Ouknine (2002), Regularization of differential equations by fractional noise. Stoc. Proc. Appl., 102, 103-116.
[7]	D. Nualart, The Malliavin Calculus and Related Topics, Springer Verlag, Berlin, Heidelberg, 2006.
[8]	Fabian A. Harang, Torstein K. Nilssen & Frank N. Proske (2022) Girsanov theorem for multifractional Brownian processes, Stochastics, 94: 8, 1137-1165, https://doi.org/10.1080/17442508.2022.2027948
[9]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications Vol. 44, Cambridge University Press, Cambridge, 1992.
[10]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Graduate Texts in Mathematics Vol. 113, Springer-Verlag, New York, 1991.
[11]	J. Lebovits and J. Levy Vehel, White noise based stochastic calculus with respect to multifractional Brownian motion, Stochastics 86 2014 87 124.
[12]	L. Martellini, Philippe Priaulet, S. Priaulet (2003). Fixed Income Securities. John Wiley
[13]	Marquez-Lago TT, Leier A, Burrage K. Anomalous diffusion and multifractional Brownian motion: simulating molecular crowding and physical obstacles in systems biology IET Systems Biology 6(4): 134-142.
[14]	Pierre R. Bertrand, Mehdi Fhima et Arnaud Guillin : Local estimation of the hurst index of multifractional brownian motion by increment ratio statistic method. Applied Mathematical Sciences, 2010.
[15]	Pierre R. Bertrand, Abdelkader Hamdouni et Samia Khadhraoui : Modelling nasdaq series by sparse multifractional brownian motion. Methodology and Computing in Applied Probability, 2010.
[16]	R. F. Peltier and J. L. Vehel. Multifractional Brownian motion : definition and preliminary results. Rapport de recherche-INRIA, 1995.
[17]	Ruili Hao, Yonghui Liu, Shoubai Wang (2013). Pricing Credit Default Swap under Fractional Vasicek Interest Rate Model. Journal of Mathematical Finance.
[18]	S. A. Stoev and M. S. Taqqu. How rich is the class of multifractional Brownian motions. Stochastic Processes and their Applications, 116: 200-221, 2006.
[19]	S. Corlay, J. Lebovits and J. L. Vehel, Multifractional stochastic volatility models, Math Financ 24 (2014) 364-402.
[20]	S. C. Lim, Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville, type, J. Phys. A: Math. Gen. 34(7) (2001), pp. 1301-1310.
[21]	Samorodnitsky, G. et M. S. Taqqu. 1994, Stable non-Gaussian random processes : stochastic models with infinite variance, vol. 1, CRC press.
[22]	S. Stoev and M. S. Taqqu. How rich is the class of multifractional Brownian motions? Stochastic Processes and their Applications, 116(2) : 200-221, 2006.
[23]	THIOUNE M., BA Demba Bocar and DIOP Bou.: Study of models without jumps with respect to fractionnal brownian motion. Journal of Computer Science and Applied Mathematics. Vol. 4, No.1, (2022), pp. 15-30. https://doi.org/10.37418/jcsam.4.1.2
[24]	X. Fernique, Integrality of Gaussian vectors, C. R. Acad. Sci. Paris Ser. A-B 270 (1970), pp. A1698-A1699.

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APA Style

EBeye, H., Thioune, M., Ba, D. B. (2026). The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods. International Journal of Theoretical and Applied Mathematics, 12(1), 31-36. https://doi.org/10.11648/j.ijtam.20261201.14

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ACS Style

EBeye, H.; Thioune, M.; Ba, D. B. The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods. Int. J. Theor. Appl. Math. 2026, 12(1), 31-36. doi: 10.11648/j.ijtam.20261201.14

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AMA Style

EBeye H, Thioune M, Ba DB. The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods. Int J Theor Appl Math. 2026;12(1):31-36. doi: 10.11648/j.ijtam.20261201.14

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@article{10.11648/j.ijtam.20261201.14,
  author = {Hamid EBeye and Moussa Thioune and Demba Bocar Ba},
  title = {The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {12},
  number = {1},
  pages = {31-36},
  doi = {10.11648/j.ijtam.20261201.14},
  url = {https://doi.org/10.11648/j.ijtam.20261201.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20261201.14},
  abstract = {To overcome some of the limitations of traditional fractional Brownian motion (fBm), multifractional Brownian motion (mBm) was created. The Holder exponent of mBm can vary along the trajectory, unlike fBm, which is advantageous for modeling processes whose regularity varies over time, like internet traffic or photos. This is the main difference between the two processes. Many continuous observations can be modeled by stochastic differential equations governed by mBm, especially in biology and finance. Using a non-stationary multifractional Brownian motion with a time-dependent Hurst parameter as the noise source and a simply linear drift coefficient, we first show the existence and uniqueness of a solution to stochastic differential equations in this article. Next, we examine multifractional Brownian motion (mBm) driven stochastic differential equation models that allow for the simulation of some discontinuity-containing phenomena.
},
 year = {2026}
}

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TY  - JOUR
T1  - The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods

AU  - Hamid EBeye
AU  - Moussa Thioune
AU  - Demba Bocar Ba
Y1  - 2026/02/04
PY  - 2026
N1  - https://doi.org/10.11648/j.ijtam.20261201.14
DO  - 10.11648/j.ijtam.20261201.14
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 31
EP  - 36
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20261201.14
AB  - To overcome some of the limitations of traditional fractional Brownian motion (fBm), multifractional Brownian motion (mBm) was created. The Holder exponent of mBm can vary along the trajectory, unlike fBm, which is advantageous for modeling processes whose regularity varies over time, like internet traffic or photos. This is the main difference between the two processes. Many continuous observations can be modeled by stochastic differential equations governed by mBm, especially in biology and finance. Using a non-stationary multifractional Brownian motion with a time-dependent Hurst parameter as the noise source and a simply linear drift coefficient, we first show the existence and uniqueness of a solution to stochastic differential equations in this article. Next, we examine multifractional Brownian motion (mBm) driven stochastic differential equation models that allow for the simulation of some discontinuity-containing phenomena.

VL  - 12
IS  - 1
ER  -

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Author Information

Hamid EBeye

Department Mathematic, UFR SET University Iba Der Thiam, Thies, Senegal

Contact Email
Moussa Thioune

Department Mathematic, UFR SET University Iba Der Thiam, Thies, Senegal

Contact Email
Demba Bocar Ba

Department Mathematic, UFR SET University Iba Der Thiam, Thies, Senegal

Contact Email

http://orcid.org/0000-0002-4303-8506

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Plain Text BibTeX RIS

APA Style

EBeye, H., Thioune, M., Ba, D. B. (2026). The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods. International Journal of Theoretical and Applied Mathematics, 12(1), 31-36. https://doi.org/10.11648/j.ijtam.20261201.14

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ACS Style

EBeye, H.; Thioune, M.; Ba, D. B. The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods. Int. J. Theor. Appl. Math. 2026, 12(1), 31-36. doi: 10.11648/j.ijtam.20261201.14

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AMA Style

EBeye H, Thioune M, Ba DB. The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods. Int J Theor Appl Math. 2026;12(1):31-36. doi: 10.11648/j.ijtam.20261201.14

Copy | Download

@article{10.11648/j.ijtam.20261201.14,
  author = {Hamid EBeye and Moussa Thioune and Demba Bocar Ba},
  title = {The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {12},
  number = {1},
  pages = {31-36},
  doi = {10.11648/j.ijtam.20261201.14},
  url = {https://doi.org/10.11648/j.ijtam.20261201.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20261201.14},
  abstract = {To overcome some of the limitations of traditional fractional Brownian motion (fBm), multifractional Brownian motion (mBm) was created. The Holder exponent of mBm can vary along the trajectory, unlike fBm, which is advantageous for modeling processes whose regularity varies over time, like internet traffic or photos. This is the main difference between the two processes. Many continuous observations can be modeled by stochastic differential equations governed by mBm, especially in biology and finance. Using a non-stationary multifractional Brownian motion with a time-dependent Hurst parameter as the noise source and a simply linear drift coefficient, we first show the existence and uniqueness of a solution to stochastic differential equations in this article. Next, we examine multifractional Brownian motion (mBm) driven stochastic differential equation models that allow for the simulation of some discontinuity-containing phenomena.
},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - The Existence and Uniqueness of SDE Solutions Diriged by Multiractional Brownian Motion , Based on Several Developed Methods

AU  - Hamid EBeye
AU  - Moussa Thioune
AU  - Demba Bocar Ba
Y1  - 2026/02/04
PY  - 2026
N1  - https://doi.org/10.11648/j.ijtam.20261201.14
DO  - 10.11648/j.ijtam.20261201.14
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 31
EP  - 36
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20261201.14
AB  - To overcome some of the limitations of traditional fractional Brownian motion (fBm), multifractional Brownian motion (mBm) was created. The Holder exponent of mBm can vary along the trajectory, unlike fBm, which is advantageous for modeling processes whose regularity varies over time, like internet traffic or photos. This is the main difference between the two processes. Many continuous observations can be modeled by stochastic differential equations governed by mBm, especially in biology and finance. Using a non-stationary multifractional Brownian motion with a time-dependent Hurst parameter as the noise source and a simply linear drift coefficient, we first show the existence and uniqueness of a solution to stochastic differential equations in this article. Next, we examine multifractional Brownian motion (mBm) driven stochastic differential equation models that allow for the simulation of some discontinuity-containing phenomena.

VL  - 12
IS  - 1
ER  -

Copy | Download