A Splitting Method for SDE Terminal-Law Estimation
Abstract
In many settings involving stochastic differential equations (SDEs), including diffusion based generative AI, our aim is to accurately generate samples from a distribution. Typically, this is done by generating i.i.d. samples of the SDE. Given a fixed simulation budget, a reasonable way to gain efficiency may be to instead generate a tree of paths through appropriately split partial paths. This suggests improved performance, but one worries about the injected dependence. In this paper, we study this issue comprehensively. We develop a general method to construct the splitting design and analyze its performance for a broad class of error measures, giving insight into the elegant underlying structure in the problem. For the Kolmogorov-Smirnov (KS) distance and maximum mean discrepancy (MMD), we identify the limiting errors of the associated empirical distributions as the simulation budget increases to infinity. We characterize a splitting strategy motivated by a corresponding asymptotic optimization problem. Overall, we observe a 12-26% improvement in mean KS over i.i.d. samples in many settings. On CIFAR-10 and FFHQ, splitting lowers the MMD by 17-26% relative to i.i.d. samples.
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BibTeX
@misc{gupta2026splitting,
title = {{A Splitting Method for SDE Terminal-Law Estimation}},
author = {Rushil Gupta and Sandeep Juneja},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12513},
eprint = {2609.12513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12513},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Splitting Method for SDE Terminal-Law Estimation
AU - Rushil Gupta
AU - Sandeep Juneja
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2609.12513
UR - https://arxiv.org/abs/2609.12513
AB - In many settings involving stochastic differential equations (SDEs), including diffusion based generative AI, our aim is to accurately generate samples from a distribution. Typically, this is done by generating i.i.d. samples of the SDE. Given a fixed simulation budget, a reasonable way to gain efficiency may be to instead generate a tree of paths through appropriately split partial paths. This suggests improved performance, but one worries about the injected dependence. In this paper, we study this issue comprehensively. We develop a general method to construct the splitting design and analyze its performance for a broad class of error measures, giving insight into the elegant underlying structure in the problem. For the Kolmogorov-Smirnov (KS) distance and maximum mean discrepancy (MMD), we identify the limiting errors of the associated empirical distributions as the simulation budget increases to infinity. We characterize a splitting strategy motivated by a corresponding asymptotic optimization problem. Overall, we observe a 12-26% improvement in mean KS over i.i.d. samples in many settings. On CIFAR-10 and FFHQ, splitting lowers the MMD by 17-26% relative to i.i.d. samples.
ER -