Randomly shifted lattice rules for SDEs with constant diffusion: error bounds uniform in time discretization and dimension
Abstract
We study randomly shifted rank-1 lattice rules for computing 𝔼[g(XT)], where XT is the terminal state of a stochastic differential equation (SDE) with constant diffusion matrix. Discretizing the SDE by the Euler--Maruyama scheme with m uniform time steps turns the expectation into an integral whose dimension grows with m and whose integrand has a nested composition structure; both complicate the lattice rule error analysis. We generate the Brownian path by the Brownian bridge construction and handle the nested structure using the multivariate Faà di Bruno formula and a rooted-tree expansion. Under suitable regularity and growth conditions, we prove that an N-point randomly shifted lattice rule with a uniformly random generating vector achieves a mean absolute sampling error of O(N−1+ε) for every ε∈(0,1/2) and every prime N, with the implied constant independent of m. To improve stability and control higher error moments, the median lattice estimator is used. The estimator takes the median of R=2k−1 independent lattice estimates and achieves an Lq error of O(N−1+ε) for every 1≤ q<k/(1−ε). We further establish error bounds independent of both the state dimension and the Brownian path dimension under an additional summability condition on the columns of the diffusion matrix. Numerical experiments show that median lattice estimators converge faster than Monte Carlo estimators and remain stable as the number of time steps m grows. For models satisfying the additional summability condition, stability is also observed as the state and Brownian path dimensions increase.
Related
- Binary dimension-free discretization and polynomial complexification on real cubes
- Shifted tapered-pole approximation on finite sectors: an exact asymptotic and dimension control
- A Shifted Balanced Polynomial Collocation Framework for Variable-Coefficient Multi-Term Time-Fractional Convection–Diffusion–Reaction Problems
- Dimension-free weak (1,1) estimates for Riesz transforms with constant drift
- Finite index constant mean curvature hypersurfaces in space forms of dimension at most seven
BibTeX
@misc{ouyang2026randomly,
title = {{Randomly shifted lattice rules for SDEs with constant diffusion: error bounds uniform in time discretization and dimension}},
author = {Du Ouyang and Josef Dick and Ziyang Ye},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07424},
eprint = {2611.07424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07424},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Randomly shifted lattice rules for SDEs with constant diffusion: error bounds uniform in time discretization and dimension
AU - Du Ouyang
AU - Josef Dick
AU - Ziyang Ye
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07424
UR - https://arxiv.org/abs/2611.07424
AB - We study randomly shifted rank-1 lattice rules for computing 𝔼[g(X_T)], where X_T is the terminal state of a stochastic differential equation (SDE) with constant diffusion matrix. Discretizing the SDE by the Euler--Maruyama scheme with m uniform time steps turns the expectation into an integral whose dimension grows with m and whose integrand has a nested composition structure; both complicate the lattice rule error analysis. We generate the Brownian path by the Brownian bridge construction and handle the nested structure using the multivariate Faà di Bruno formula and a rooted-tree expansion. Under suitable regularity and growth conditions, we prove that an N-point randomly shifted lattice rule with a uniformly random generating vector achieves a mean absolute sampling error of O(N^−1+ε) for every ε∈(0,1/2) and every prime N, with the implied constant independent of m. To improve stability and control higher error moments, the median lattice estimator is used. The estimator takes the median of R=2k−1 independent lattice estimates and achieves an L^q error of O(N^−1+ε) for every 1≤ q<k/(1−ε). We further establish error bounds independent of both the state dimension and the Brownian path dimension under an additional summability condition on the columns of the diffusion matrix. Numerical experiments show that median lattice estimators converge faster than Monte Carlo estimators and remain stable as the number of time steps m grows. For models satisfying the additional summability condition, stability is also observed as the state and Brownian path dimensions increase.
ER -