Neural Drift Estimation for Ergodic Diffusions: Non-parametric Analysis and Numerical Exploration

TL;DR

This paper develops a non-parametric neural network-based method for estimating the drift in ergodic stochastic differential equations, providing theoretical bounds and practical inference techniques for noisy data.

Contribution

It introduces a novel approach combining neural networks with ergodic SDE drift estimation, supported by theoretical bounds and practical inference methods.

Findings

Effective drift estimation on simulated data

Theoretical generalization bounds established

Practical inference method for noisy functional data

Abstract

We take into consideration generalization bounds for the problem of the estimation of the drift component for ergodic stochastic differential equations, when the estimator is a ReLU neural network and the estimation is non-parametric with respect to the statistical model. We show a practical way to enforce the theoretical estimation procedure, enabling inference on noisy and rough functional data. Results are shown for a simulated It\^o-Taylor approximation of the sample paths.