Higher-order approximations for the denoising target in drift estimation

Extend the denoising-based drift estimator by incorporating higher-order approximations of the conditional denoising target E[X_0 | X_τ, Y] beyond the first-order Euler–Maruyama approximation used to derive the estimator, and analyze the resulting estimator’s properties under this refinement.

Background

The proposed estimator recovers the drift by first training a denoiser via a conditional diffusion objective and then using a first-order (Euler–Maruyama) approximation of the increment distribution to relate the learned denoising target to the drift. This approximation facilitates a closed-form estimator but may introduce bias at smaller sampling intervals or in more complex dynamics.

The authors explicitly identify extending the estimator to incorporate higher-order approximations of the denoising target as an open direction, suggesting potential improvements in accuracy and robustness that have not yet been developed or analyzed.