Drift Estimation for Diffusion Processes Using Neural Networks Based on Discretely Observed Independent Paths

TL;DR

This paper introduces a neural network estimator for nonparametric drift function estimation in diffusion processes, achieving improved convergence rates and dimension-independent performance in numerical experiments.

Contribution

It proposes a neural network-based method with explicit convergence rates for drift estimation from discretely observed diffusion paths, outperforming traditional spline methods.

Findings

Neural network estimator has a convergence rate scaling as log N / N.

The method effectively captures local features and performs well in high dimensions.

Empirical results show improved convergence over spline methods, especially in higher dimensions.

Abstract

This paper addresses the nonparametric estimation of the drift function over a compact domain for a time-homogeneous diffusion process, based on high-frequency discrete observations from $N$ independent trajectories. We propose a neural network-based estimator and derive a non-asymptotic convergence rate, decomposed into a training error, an approximation error, and a diffusion-related term scaling as ${\log N}/{N}$. For compositional drift functions, we establish an explicit rate. In the numerical experiments, we consider a drift function with local fluctuations generated by a double-layer compositional structure featuring local oscillations, and show that the empirical convergence rate becomes independent of the input dimension $d$. Compared to the $B$-spline method, the neural network estimator achieves better convergence rates and more effectively captures local features, particularly in higher-dimensional settings.