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

2025-11-14

AI Summary

This paper addresses the nonparametric estimation of the drift function for a time-homogeneous diffusion process on a compact domain, based on high-frequency discrete observations from $N$ independent sample paths. We propose a neural network–driven estimator that explicitly characterizes the convergence rate for drift functions with composite structures; the total estimation error is decomposed into training error, approximation error, and a diffusion-related term. Theoretically, we establish a dimension-free convergence rate—i.e., the rate does not deteriorate with increasing input dimension $d$—thereby circumventing the “curse of dimensionality.” Leveraging the strong expressive power of deep neural networks, our method accurately captures local oscillatory features of the drift function. Numerical experiments demonstrate that, compared to classical approaches such as B-splines, the proposed method achieves faster convergence, superior robustness in high dimensions, and significantly improved estimation accuracy and generalization performance.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• Estimating drift functions for diffusion processes from discrete observations
• Developing neural network estimators with non-asymptotic convergence rates
• Capturing local features in high-dimensional compositional drift functions

Innovation

• Neural network estimator for drift function estimation
• Non-asymptotic convergence rate with three components
• Better performance than B-spline in high dimensions