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.

References

Several important questions remain open. These include developing a theoretical understanding for how different noise schedules affect the bias-variance properties of the estimator under specific architectures, as well as extending the estimator construction to incorporate higher order approximations of the denoising target.

— Drift Estimation for Stochastic Differential Equations with Denoising Diffusion Models  (2602.17830 - Costa et al., 19 Feb 2026) in Conclusion

Whether the first order update which gives exact weights and no extrapolations are needed or the second order update giving approximate weights and the extrapolation afterwards is cheaper needs to be investigated for the particular application one has.

— Accurate Sampling from Diffusion Models  (2609.29902 - Sexty, 24 Sep 2026) in Section 5, Simulation setup, discussion following equations (\ref{detexpand}) and (\ref{traceapprox})