Title: Establishing Error Bounds for a Diffusion Model-Driven Drift Estimator

Abstract:

Estimating parameters within stochastic differential equations constitutes a foundational statistical challenge with significant implications across numerous scientific disciplines. While Tapia Costa et al. (2026) recently proposed an innovative approach for drift estimation—assuming the diffusion parameter is known and utilizing discrete samples from multiple trajectories—the theoretical underpinnings of this method remained unverified. Their technique reframes drift estimation as a denoising task, drawing upon methodologies from (conditional) score-matching diffusion models. Although empirical evaluations demonstrated favorable outcomes across various drift categories, the absence of theoretical guarantees for the estimator’s performance was a notable omission. This note bridges that theoretical gap by applying concepts from diffusion model theory. Specifically, we establish a concrete risk bound for the time-averaged mean-squared error of the proposed drift estimator. This bound dissects the total risk into four distinct components: (i) Euler-Maruyama discretization errors, (ii) approximations in the score or denoiser, (iii) noise initialization effects, and (iv) sampling variance. By isolating these factors, our analysis clarifies the intricate trade-offs among various hyperparameters and error sources inherent in the estimator.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC