A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management

Bou Diop

doi:doi:10.11648/j.ajam.20261401.12

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A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management

Bou Diop^*

Published in American Journal of Applied Mathematics (Volume 14, Issue 1)

Received: 25 October 2025 Accepted: 4 November 2025 Published: 15 January 2026

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Abstract

This research develops an innovative mathematical framework that unifies classical and modern approaches to stochastic differential equations (SDEs) driven by irregular paths. We introduce a novel Newton-Cotes integration method that bridges Young integration and rough path theory, providing comprehensive solutions for processes with Hölder continuous sample paths. The theoretical foundation establishes existence, uniqueness, and regularity results across the entire roughness spectrum. Our methodology offers practical advantages through adaptive numerical schemes with proven convergence rates and robust parameter estimation techniques combining maximum likelihood and Bayesian approaches. The framework’s real-world utility is demonstrated through a detailed case study of groundwater management in Senegal, where our model achieves a 52%improvement in prediction accuracy over traditional methods. This enhancement enables more reliable drought early warnings and sustainable water resource planning in semi-arid regions facing climate uncertainty. The unified approach has broad applicability across scientific domains dealing with irregular data patterns, including finance, environmental science, and engineering.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 1)
DOI	10.11648/j.ajam.20261401.12
Page(s)	10-13
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

Stochastic Differential Equations, Hèlder Continuity, Rough Path Theory, Newton-Cotes Integration, Groundwater Management, Parameter Estimation, Senegal Hydrology, Water Resources

References

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[2]	P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511845079
[3]	Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, 2008. https://doi.org/10.1007/978-3-540-75873-0
[4]	F. Biagini, Y. Hu, B. Ãksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, 2008. https://doi.org/10.1007/978-1-84628-797-8
[5]	B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, no. 4, pp. 422-437, 1968. https://doi.org/10.1137/1010093
[6]	M. Couceiro et al., “Rough path methods for hydrological time series,” Water Resources Research, vol. 57, no. 4, p. e2020WR028463, 2021. https://doi.org/10.1029/2020WR028463
[7]	M. Gubinelli, “Controlling rough paths,” Journal of Functional Analysis, vol. 216, no. 1, pp. 86-140, 2004. https://doi.org/10.1016/j.jfa.2004.01.002
[8]	L. C. Young, “An inequality of the Hèlder type, connected with Stieltjes integration,” Acta Mathematica, vol. 67, no. 1, pp. 251-282, 1936. https://doi.org/10.1007/BF02401743
[9]	H. Doss, “Liens entre équations différentielles stochastiques et ordinaires,” Annales de l’Institut Henri Poincaré Section B, vol. 13, no. 2, pp. 99-125, 1977. https://doi.org/10.24033/asens.1326
[10]	H. J. Sussman, “On the gap between deterministic and stochastic ordinary differential equations,” The Annals of Probability, vol. 6, no. 1, pp. 19-41, 1978. https://doi.org/10.1214/aop/1176995608
[11]	M. K. Sene et al., “Groundwater sustainability under climate change in Senegal,” Environmental Research Letters, vol. 18, no. 3, p. 034025, 2023. https://doi.org/ 10.1088/1748-9326/acb5e2
[12]	A. Neuenkirch and I. Nourdin, “Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion,” Journal of Theoretical Probability, vol. 21, no. 4, pp. 729-753, 2008. https://doi.org/10.1007/s10959-007-0123-9
[13]	C. Li and X. Wang, “High-order numerical schemes for rough differential equations,” SIAM Journal on Numerical Analysis, vol. 58, no. 2, pp. 1024-1050, 2020. https://doi.org/10.1137/19M125935X
[14]	B. L. S. Prakasa Rao, Statistical Inference for Fractional Diffusion Processes. Wiley, 2012. https://doi.org/10.1002/9781119945673
[15]	Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes. Springer, 2004. https://doi.org/10.1007/978-1-4471-3866-2
[16]	S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, 1993. https://doi.org/10.1201/9781003410667

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Diop, B. (2026). A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management. American Journal of Applied Mathematics, 14(1), 10-13. https://doi.org/10.11648/j.ajam.20261401.12

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ACS Style

Diop, B. A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management. Am. J. Appl. Math. 2026, 14(1), 10-13. doi: 10.11648/j.ajam.20261401.12

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AMA Style

Diop B. A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management. Am J Appl Math. 2026;14(1):10-13. doi: 10.11648/j.ajam.20261401.12

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@article{10.11648/j.ajam.20261401.12,
  author = {Bou Diop},
  title = {A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management
},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {1},
  pages = {10-13},
  doi = {10.11648/j.ajam.20261401.12},
  url = {https://doi.org/10.11648/j.ajam.20261401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261401.12},
  abstract = {This research develops an innovative mathematical framework that unifies classical and modern approaches to stochastic differential equations (SDEs) driven by irregular paths. We introduce a novel Newton-Cotes integration method that bridges Young integration and rough path theory, providing comprehensive solutions for processes with Hölder continuous sample paths. The theoretical foundation establishes existence, uniqueness, and regularity results across the entire roughness spectrum. Our methodology offers practical advantages through adaptive numerical schemes with proven convergence rates and robust parameter estimation techniques combining maximum likelihood and Bayesian approaches. The framework’s real-world utility is demonstrated through a detailed case study of groundwater management in Senegal, where our model achieves a 52%improvement in prediction accuracy over traditional methods. This enhancement enables more reliable drought early warnings and sustainable water resource planning in semi-arid regions facing climate uncertainty. The unified approach has broad applicability across scientific domains dealing with irregular data patterns, including finance, environmental science, and engineering.
},
 year = {2026}
}

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TY  - JOUR
T1  - A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management

AU  - Bou Diop
Y1  - 2026/01/15
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261401.12
DO  - 10.11648/j.ajam.20261401.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
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PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261401.12
AB  - This research develops an innovative mathematical framework that unifies classical and modern approaches to stochastic differential equations (SDEs) driven by irregular paths. We introduce a novel Newton-Cotes integration method that bridges Young integration and rough path theory, providing comprehensive solutions for processes with Hölder continuous sample paths. The theoretical foundation establishes existence, uniqueness, and regularity results across the entire roughness spectrum. Our methodology offers practical advantages through adaptive numerical schemes with proven convergence rates and robust parameter estimation techniques combining maximum likelihood and Bayesian approaches. The framework’s real-world utility is demonstrated through a detailed case study of groundwater management in Senegal, where our model achieves a 52%improvement in prediction accuracy over traditional methods. This enhancement enables more reliable drought early warnings and sustainable water resource planning in semi-arid regions facing climate uncertainty. The unified approach has broad applicability across scientific domains dealing with irregular data patterns, including finance, environmental science, and engineering.

VL  - 14
IS  - 1
ER  -

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Author Information

Bou Diop

Department of Mathematics, IBA DER THIAM University, Thiès, Senegal

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APA Style

Diop, B. (2026). A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management. American Journal of Applied Mathematics, 14(1), 10-13. https://doi.org/10.11648/j.ajam.20261401.12

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ACS Style

Diop, B. A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management. Am. J. Appl. Math. 2026, 14(1), 10-13. doi: 10.11648/j.ajam.20261401.12

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AMA Style

Diop B. A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management. Am J Appl Math. 2026;14(1):10-13. doi: 10.11648/j.ajam.20261401.12

Copy | Download

@article{10.11648/j.ajam.20261401.12,
  author = {Bou Diop},
  title = {A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management
},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {1},
  pages = {10-13},
  doi = {10.11648/j.ajam.20261401.12},
  url = {https://doi.org/10.11648/j.ajam.20261401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261401.12},
  abstract = {This research develops an innovative mathematical framework that unifies classical and modern approaches to stochastic differential equations (SDEs) driven by irregular paths. We introduce a novel Newton-Cotes integration method that bridges Young integration and rough path theory, providing comprehensive solutions for processes with Hölder continuous sample paths. The theoretical foundation establishes existence, uniqueness, and regularity results across the entire roughness spectrum. Our methodology offers practical advantages through adaptive numerical schemes with proven convergence rates and robust parameter estimation techniques combining maximum likelihood and Bayesian approaches. The framework’s real-world utility is demonstrated through a detailed case study of groundwater management in Senegal, where our model achieves a 52%improvement in prediction accuracy over traditional methods. This enhancement enables more reliable drought early warnings and sustainable water resource planning in semi-arid regions facing climate uncertainty. The unified approach has broad applicability across scientific domains dealing with irregular data patterns, including finance, environmental science, and engineering.
},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - A Unified Framework for Stochastic Differential Equations Driven by H ö lder Continuous Functions with Applications to Senegalese Groundwater Management

AU  - Bou Diop
Y1  - 2026/01/15
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261401.12
DO  - 10.11648/j.ajam.20261401.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 10
EP  - 13
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261401.12
AB  - This research develops an innovative mathematical framework that unifies classical and modern approaches to stochastic differential equations (SDEs) driven by irregular paths. We introduce a novel Newton-Cotes integration method that bridges Young integration and rough path theory, providing comprehensive solutions for processes with Hölder continuous sample paths. The theoretical foundation establishes existence, uniqueness, and regularity results across the entire roughness spectrum. Our methodology offers practical advantages through adaptive numerical schemes with proven convergence rates and robust parameter estimation techniques combining maximum likelihood and Bayesian approaches. The framework’s real-world utility is demonstrated through a detailed case study of groundwater management in Senegal, where our model achieves a 52%improvement in prediction accuracy over traditional methods. This enhancement enables more reliable drought early warnings and sustainable water resource planning in semi-arid regions facing climate uncertainty. The unified approach has broad applicability across scientific domains dealing with irregular data patterns, including finance, environmental science, and engineering.

VL  - 14
IS  - 1
ER  -

Copy | Download