VEST: Variational Drift Estimation from Noisy Observations

Nonparametric drift estimation for stochastic differential equations from multiple independent trajectories with additive observation noise remains challenging in statistics and machine learning. However, direct regression on noisy increments amplifies measurement error, whereas regression on smoothed trajectories treated as exact discards uncertainty and dependence between consecutive latent states. Thus, we propose Variational Drift Estimation from Noisy Observations (VEST), a method that alternates between Gaussian Markov smoothing and neural drift learning. An iterated extended Kalman smoother proposes updates to the Gaussian approximation, while the drift is updated by regressing on conditional targets derived from its adjacent state moments. An acceptance rule based on a shared variational objective determines whether each proposed update is accepted. Moreover, we give theoretical guarantees on the discretization error of the variational objective and the convergence of the algorithm. Experiments show that VEST outperforms existing related methods.