Statistical Spatially Inhomogeneous Diffusion Inference

Inferring a diffusion equation from discretely-observed measurements is a statistical challenge of significant importance in a variety of fields, from single-molecule tracking in biophysical systems to modeling financial instruments. Assuming that the underlying dynamical process obeys a d-dimensional stochastic differential equation of the form dxt = b(xt)dt + Σ(xt)dwt, we propose neural network-based estimators of both the drift b and the spatially-inhomogeneous diffusion tensor D = ΣΣ^T /2 and provide statistical convergence guarantees when b and D are s-Hölder continuous. Notably, our bound aligns 2s with the minimax optimal rate N^-2s+d for nonparametric function estimation even in the presence of correlation within observational data, which necessitates careful handling when establishing fast-rate generalization bounds. Our theoretical results are bolstered by numerical experiments demonstrating accurate inference of spatially-inhomogeneous diffusion tensors.