Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs

Bou Diop

doi:doi:10.11648/j.sjams.20261404.12

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Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs

Bou Diop^*

Published in Science Journal of Applied Mathematics and Statistics (Volume 14, Issue 4)

Received: 19 December 2025 Accepted: 21 January 2026 Published: 20 August 2026

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Abstract

This paper develops a comprehensive framework for stochastic integration with respect to multidimensional fractional Brownian sheets, with particular emphasis on the anisotropic setting where each Hurst index exceeds one-half. We introduce a regularizationbased approach that accounts for the directional structure of the sheet and establish its equivalence to the Malliavin calculus construction. Building on this foundation, we obtain wellposedness and regularity results for a class of stochastic partial differential equations driven by such sheets. The regularization method provides an intuitive interpretation of the integral while preserving the directional heterogeneity inherent in anisotropic models. We prove that the regularized integral converges in the mean-square sense to the Skorokhod integral, establishing a duality relation that connects our constructive approach to established analytical frameworks. Our well-posedness results are obtained through a fixed-point argument in suitably chosen solution spaces, while the regularity analysis relies on anisotropic versions of the classical Kolmogorov criterion. Notably, the H¨older exponents we obtain directly reflect the directional smoothness of the driving sheet. This feature has practical implications for numerical discretization and statistical estimation. Our findings extend known one-dimensional theories to genuinely multidimensional and anisotropic regimes, offering new tools for modelling systems with long-range dependence and directional heterogeneity. Applications include fluid flows with directional turbulence, financial markets with correlated assets exhibiting different memory properties, and biological tissues with anisotropic diffusion characteristics.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 14, Issue 4)
DOI	10.11648/j.sjams.20261404.12
Page(s)	106-112
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 Partial Differential Equations, Fractional Brownian Sheets, Anisotropic Regularity, Malliavin Calculus, Stochastic Integration, Regularization Method, H¨older Continuity, Long-Range Dependence

References

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APA Style

Diop, B. (2026). Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs. Science Journal of Applied Mathematics and Statistics, 14(4), 106-112. https://doi.org/10.11648/j.sjams.20261404.12

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ACS Style

Diop, B. Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs. Sci. J. Appl. Math. Stat. 2026, 14(4), 106-112. doi: 10.11648/j.sjams.20261404.12

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AMA Style

Diop B. Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs. Sci J Appl Math Stat. 2026;14(4):106-112. doi: 10.11648/j.sjams.20261404.12

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@article{10.11648/j.sjams.20261404.12,
  author = {Bou Diop},
  title = {Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {14},
  number = {4},
  pages = {106-112},
  doi = {10.11648/j.sjams.20261404.12},
  url = {https://doi.org/10.11648/j.sjams.20261404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20261404.12},
  abstract = {This paper develops a comprehensive framework for stochastic integration with respect to multidimensional fractional Brownian sheets, with particular emphasis on the anisotropic setting where each Hurst index exceeds one-half. We introduce a regularizationbased approach that accounts for the directional structure of the sheet and establish its equivalence to the Malliavin calculus construction. Building on this foundation, we obtain wellposedness and regularity results for a class of stochastic partial differential equations driven by such sheets. The regularization method provides an intuitive interpretation of the integral while preserving the directional heterogeneity inherent in anisotropic models. We prove that the regularized integral converges in the mean-square sense to the Skorokhod integral, establishing a duality relation that connects our constructive approach to established analytical frameworks. Our well-posedness results are obtained through a fixed-point argument in suitably chosen solution spaces, while the regularity analysis relies on anisotropic versions of the classical Kolmogorov criterion. Notably, the H¨older exponents we obtain directly reflect the directional smoothness of the driving sheet. This feature has practical implications for numerical discretization and statistical estimation. Our findings extend known one-dimensional theories to genuinely multidimensional and anisotropic regimes, offering new tools for modelling systems with long-range dependence and directional heterogeneity. Applications include fluid flows with directional turbulence, financial markets with correlated assets exhibiting different memory properties, and biological tissues with anisotropic diffusion characteristics.},
 year = {2026}
}

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TY  - JOUR
T1  - Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs
AU  - Bou Diop
Y1  - 2026/08/20
PY  - 2026
N1  - https://doi.org/10.11648/j.sjams.20261404.12
DO  - 10.11648/j.sjams.20261404.12
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 106
EP  - 112
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20261404.12
AB  - This paper develops a comprehensive framework for stochastic integration with respect to multidimensional fractional Brownian sheets, with particular emphasis on the anisotropic setting where each Hurst index exceeds one-half. We introduce a regularizationbased approach that accounts for the directional structure of the sheet and establish its equivalence to the Malliavin calculus construction. Building on this foundation, we obtain wellposedness and regularity results for a class of stochastic partial differential equations driven by such sheets. The regularization method provides an intuitive interpretation of the integral while preserving the directional heterogeneity inherent in anisotropic models. We prove that the regularized integral converges in the mean-square sense to the Skorokhod integral, establishing a duality relation that connects our constructive approach to established analytical frameworks. Our well-posedness results are obtained through a fixed-point argument in suitably chosen solution spaces, while the regularity analysis relies on anisotropic versions of the classical Kolmogorov criterion. Notably, the H¨older exponents we obtain directly reflect the directional smoothness of the driving sheet. This feature has practical implications for numerical discretization and statistical estimation. Our findings extend known one-dimensional theories to genuinely multidimensional and anisotropic regimes, offering new tools for modelling systems with long-range dependence and directional heterogeneity. Applications include fluid flows with directional turbulence, financial markets with correlated assets exhibiting different memory properties, and biological tissues with anisotropic diffusion characteristics.
VL  - 14
IS  - 4
ER  -

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Author Information

Bou Diop

Department of Mathematics, University of Thies, Thies, Senegal

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http://orcid.org/0009-0009-2741-8953

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APA Style

Diop, B. (2026). Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs. Science Journal of Applied Mathematics and Statistics, 14(4), 106-112. https://doi.org/10.11648/j.sjams.20261404.12

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ACS Style

Diop, B. Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs. Sci. J. Appl. Math. Stat. 2026, 14(4), 106-112. doi: 10.11648/j.sjams.20261404.12

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AMA Style

Diop B. Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs. Sci J Appl Math Stat. 2026;14(4):106-112. doi: 10.11648/j.sjams.20261404.12

Copy | Download

@article{10.11648/j.sjams.20261404.12,
  author = {Bou Diop},
  title = {Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {14},
  number = {4},
  pages = {106-112},
  doi = {10.11648/j.sjams.20261404.12},
  url = {https://doi.org/10.11648/j.sjams.20261404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20261404.12},
  abstract = {This paper develops a comprehensive framework for stochastic integration with respect to multidimensional fractional Brownian sheets, with particular emphasis on the anisotropic setting where each Hurst index exceeds one-half. We introduce a regularizationbased approach that accounts for the directional structure of the sheet and establish its equivalence to the Malliavin calculus construction. Building on this foundation, we obtain wellposedness and regularity results for a class of stochastic partial differential equations driven by such sheets. The regularization method provides an intuitive interpretation of the integral while preserving the directional heterogeneity inherent in anisotropic models. We prove that the regularized integral converges in the mean-square sense to the Skorokhod integral, establishing a duality relation that connects our constructive approach to established analytical frameworks. Our well-posedness results are obtained through a fixed-point argument in suitably chosen solution spaces, while the regularity analysis relies on anisotropic versions of the classical Kolmogorov criterion. Notably, the H¨older exponents we obtain directly reflect the directional smoothness of the driving sheet. This feature has practical implications for numerical discretization and statistical estimation. Our findings extend known one-dimensional theories to genuinely multidimensional and anisotropic regimes, offering new tools for modelling systems with long-range dependence and directional heterogeneity. Applications include fluid flows with directional turbulence, financial markets with correlated assets exhibiting different memory properties, and biological tissues with anisotropic diffusion characteristics.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Stochastic Integration with Respect to Fractional Brownian Sheets and Applications to Anisotropic SPDEs
AU  - Bou Diop
Y1  - 2026/08/20
PY  - 2026
N1  - https://doi.org/10.11648/j.sjams.20261404.12
DO  - 10.11648/j.sjams.20261404.12
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 106
EP  - 112
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20261404.12
AB  - This paper develops a comprehensive framework for stochastic integration with respect to multidimensional fractional Brownian sheets, with particular emphasis on the anisotropic setting where each Hurst index exceeds one-half. We introduce a regularizationbased approach that accounts for the directional structure of the sheet and establish its equivalence to the Malliavin calculus construction. Building on this foundation, we obtain wellposedness and regularity results for a class of stochastic partial differential equations driven by such sheets. The regularization method provides an intuitive interpretation of the integral while preserving the directional heterogeneity inherent in anisotropic models. We prove that the regularized integral converges in the mean-square sense to the Skorokhod integral, establishing a duality relation that connects our constructive approach to established analytical frameworks. Our well-posedness results are obtained through a fixed-point argument in suitably chosen solution spaces, while the regularity analysis relies on anisotropic versions of the classical Kolmogorov criterion. Notably, the H¨older exponents we obtain directly reflect the directional smoothness of the driving sheet. This feature has practical implications for numerical discretization and statistical estimation. Our findings extend known one-dimensional theories to genuinely multidimensional and anisotropic regimes, offering new tools for modelling systems with long-range dependence and directional heterogeneity. Applications include fluid flows with directional turbulence, financial markets with correlated assets exhibiting different memory properties, and biological tissues with anisotropic diffusion characteristics.
VL  - 14
IS  - 4
ER  -

Copy | Download