A Splitting Method for SDE Terminal-Law Estimation

• URL: http://arxiv.org/abs/2609.12513v1
• Date: Fri, 11 Sep 2026 07:20:11 GMT
• Title: A Splitting Method for SDE Terminal-Law Estimation
• Abstract summary: 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.<n>We identify the limiting errors of the empirical distributions as the simulation budget increases to infinity.<n>We observe a 10-25% improvement in mean error over i.i.d. samples in many settings.
• Score: 1.6440434996206623
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In many settings involving stochastic differential equations, including in diffusion based generative AI, our aim is to accurately generate samples from a terminal distribution. Typically, this is done by generating i.i.d. samples of diffusion paths. 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. With Kolmogorov-Smirnov distance as a measure of accuracy, 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. The theoretical results bring out the elegant underlying structure in the problem. Practical implementation involves two phases, an initial estimation phase and a final inference phase. Overall, we observe a 10-25% improvement in mean error over i.i.d. samples in many settings. In an exploratory CIFAR-10 study, our method reduces the maximum mean discrepancy by 8-13%.

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