Quantitative asymptotic error analysis and Richardson extrapolation for exponential Euler approximations of stochastic wave equations
Abstract
We establish quantitative weak convergence for the joint terminal law of the solution to a stochastic wave equation and the normalized error of its exponential Euler approximation. For multiplicative trace-class noise and spatial regularity α∈(1,2], the normalized error converges to its limiting distribution at rate O(m−(α−1)), where m is the number of time steps. This estimate identifies the leading weak bias and yields a first-order weak-error expansion with remainder O(m−α). Moreover, by canceling the leading weak-error term, we construct a Richardson extrapolation scheme with weak convergence order α for α∈(1,2] and show that this rate is sharp. To overcome the lack of parabolic smoothing and control the remainders amplified by normalization, we combine dimension-uniform regularity estimates for the joint Kolmogorov equation with a conditional weak expansion and backward transport of coefficient directions.
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BibTeX
@misc{cui2026quantitative,
title = {{Quantitative asymptotic error analysis and Richardson extrapolation for exponential Euler approximations of stochastic wave equations}},
author = {Jianbo Cui and Diancong Jin and Huagui Liu and Derui Sheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16896},
eprint = {2610.16896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16896},
}
RIS
TY - UNPB
TI - Quantitative asymptotic error analysis and Richardson extrapolation for exponential Euler approximations of stochastic wave equations
AU - Jianbo Cui
AU - Diancong Jin
AU - Huagui Liu
AU - Derui Sheng
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16896
UR - https://arxiv.org/abs/2610.16896
AB - We establish quantitative weak convergence for the joint terminal law of the solution to a stochastic wave equation and the normalized error of its exponential Euler approximation. For multiplicative trace-class noise and spatial regularity α∈(1,2], the normalized error converges to its limiting distribution at rate O(m^−(α−1)), where m is the number of time steps. This estimate identifies the leading weak bias and yields a first-order weak-error expansion with remainder O(m^−α). Moreover, by canceling the leading weak-error term, we construct a Richardson extrapolation scheme with weak convergence order α for α∈(1,2] and show that this rate is sharp. To overcome the lack of parabolic smoothing and control the remainders amplified by normalization, we combine dimension-uniform regularity estimates for the joint Kolmogorov equation with a conditional weak expansion and backward transport of coefficient directions.
ER -