Statistical Spatially Inhomogeneous Diffusion Inference

Abstract

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 we propose neural network-based estimators of both the drift b and the spatially-inhomogeneous diffusion tensor D = ^T and provide statistical convergence guarantees when b and D are s-H\"older continuous. Notably, our bound aligns with the minimax optimal rate N^-2s2s+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.

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Reproductions