Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions
Abstract
This paper studies the strong convergence order of structure-preserving stochastic theta Milstein methods for a class of index-$1$ stochastic differential algebraic equations (SDAEs) with time-dependent singular matrices and non-globally Lipschitz coefficients. The singular matrix is allowed to vary in time while preserving a fixed differential algebraic splitting, and the drift and diffusion coefficients may exhibit superlinear growth. By exploiting the index-$1$ algebraic-differential decomposition of the exact solution, we identify the Milstein coefficient of the reduced stochastic differential equation directly in the original SDAE variables and establish the well-posedness and constraint preserving property of the proposed method for θ∈[1/2,1]. Under a coupled monotonicity condition and suitable polynomial regularity assumptions, the method is proved to preserve the algebraic constraints at all time levels and to converge with strong order one in the root mean square norm. Numerical experiments confirm the structure-preserving property and the theoretical convergence order.
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BibTeX
@misc{zhu2026strong,
title = {{Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions}},
author = {Caiyuan Zhu and Shiye Wan and Ziheng Chen and Lin Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14583},
eprint = {2609.14583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14583},
}
RIS
TY - UNPB
TI - Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions
AU - Caiyuan Zhu
AU - Shiye Wan
AU - Ziheng Chen
AU - Lin Chen
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14583
UR - https://arxiv.org/abs/2609.14583
AB - This paper studies the strong convergence order of structure-preserving stochastic theta Milstein methods for a class of index-$1$ stochastic differential algebraic equations (SDAEs) with time-dependent singular matrices and non-globally Lipschitz coefficients. The singular matrix is allowed to vary in time while preserving a fixed differential algebraic splitting, and the drift and diffusion coefficients may exhibit superlinear growth. By exploiting the index-$1$ algebraic-differential decomposition of the exact solution, we identify the Milstein coefficient of the reduced stochastic differential equation directly in the original SDAE variables and establish the well-posedness and constraint preserving property of the proposed method for θ∈[1/2,1]. Under a coupled monotonicity condition and suitable polynomial regularity assumptions, the method is proved to preserve the algebraic constraints at all time levels and to converge with strong order one in the root mean square norm. Numerical experiments confirm the structure-preserving property and the theoretical convergence order.
ER -