Error Bounds for a Diffusion Model-Based Drift Estimator

2026-06-01

AI Summary

This work addresses the problem of estimating the drift function in stochastic differential equations when the diffusion coefficient is known, framing it as a denoising task amenable to diffusion-based modeling. By leveraging conditional score matching, the method recovers the drift function from discrete observations across multiple sample trajectories. The study establishes, for the first time, an explicit time-averaged mean squared error risk bound for this class of estimators. The theoretical analysis elucidates the interplay among four key sources of error: Euler–Maruyama discretization, score approximation, noise initialization, and sampling variance. The resulting risk decomposition provides a sharp characterization of how various hyperparameters influence estimation accuracy, thereby offering rigorous theoretical guarantees for drift estimation powered by diffusion models.

Abstract

Parameter estimation in stochastic differential equations is a classical statistical problem of much importance in many scientific fields. Recent work of Tapia Costa et al. (2026) introduced a novel technique for estimating the drift when the diffusion parameter is known, using discrete samples from multiple trajectories. Their method treats drift estimation as a denoising problem, and leverages tools from (conditional) score-matching diffusion models. Although their experiments showed promising results across different drift classes, the question of theoretical guarantees for their estimator was left unanswered. In this note, we address this gap by exploiting techniques from diffusion model theory. More concretely, we derive an explicit risk bound for the time-averaged mean-squared error of said drift estimator. Our bound decomposes the risk into the (i) Euler-Maruyama discretization, (ii) score/denoiser approximation, (iii) noise initialization, and (iv) sampling variance, revealing the trade-offs between the different hyperparameters and sources of error in the estimator.

Problem

• drift estimation
• diffusion models
• error bounds
• stochastic differential equations
• score matching

Innovation

• diffusion models
• drift estimation
• error bounds
• score matching
• stochastic differential equations