Stratonovich SDEs in High Dimensions: Weak Convergence Approach to Generalized Trivial Coupling
- 1Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
- 2Department of Statistics and Operations Research, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Abstract
This paper uses a generalized trivial coupling method to study the weak convergence of Stratonovich stochastic differential equations (SDEs) in high-dimensional environments. In contrast to standard methods that depend on the invertibility of the diffusion matrix or are restricted to low dimensions, our methodology deals with degenerate and non-invertible systems in arbitrary dimensions. We develop a theoretical framework for weak approximation, establishing the weak order of convergence under standard regularity assumptions. The method is integrated with multi-level Monte Carlo (MLMC) techniques to achieve optimal computational complexity O(ϵ−2) for mean square error ϵ2. We provide a detailed analysis of the variance decay, proving that Vℓ=O(δℓ2) for the coupled estimator, which leads to optimal complexity. Representative numerical tests on a 4-dimensional non-invertible system validate the theoretical results, proving the usefulness of the suggested approach. The numerical results demonstrate the computing benefits of the multi-level technique in high dimensions and validate the theoretical convergence rates. By offering both theoretical insights and useful computational methods for high-dimensional stochastic systems, this work advances the expanding field of computational stochastic analysis.
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BibTeX
@article{alnafisah2026stratonovich,
title = {{Stratonovich SDEs in High Dimensions: Weak Convergence Approach to Generalized Trivial Coupling}},
author = {Yousef Alnafisah and M. A. Sohaly},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203760},
url = {https://www.mdpi.com/2227-7390/14/20/3760},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Stratonovich SDEs in High Dimensions: Weak Convergence Approach to Generalized Trivial Coupling
AU - Yousef Alnafisah
AU - M. A. Sohaly
PY - 2026
DA - 2026/10/16
JO - Mathematics
DO - 10.3390/math14203760
UR - https://www.mdpi.com/2227-7390/14/20/3760
AB - This paper uses a generalized trivial coupling method to study the weak convergence of Stratonovich stochastic differential equations (SDEs) in high-dimensional environments. In contrast to standard methods that depend on the invertibility of the diffusion matrix or are restricted to low dimensions, our methodology deals with degenerate and non-invertible systems in arbitrary dimensions. We develop a theoretical framework for weak approximation, establishing the weak order of convergence under standard regularity assumptions. The method is integrated with multi-level Monte Carlo (MLMC) techniques to achieve optimal computational complexity O(ϵ−2) for mean square error ϵ2. We provide a detailed analysis of the variance decay, proving that Vℓ=O(δℓ2) for the coupled estimator, which leads to optimal complexity. Representative numerical tests on a 4-dimensional non-invertible system validate the theoretical results, proving the usefulness of the suggested approach. The numerical results demonstrate the computing benefits of the multi-level technique in high dimensions and validate the theoretical convergence rates. By offering both theoretical insights and useful computational methods for high-dimensional stochastic systems, this work advances the expanding field of computational stochastic analysis.
SN - 2227-7390
ER -