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

TL;DR

This work tackles nonparametric drift estimation for diffusion processes from high-frequency, discretely observed independent paths by introducing a neural-network based estimator that operates component-wise on a compact domain. It derives a non-asymptotic risk bound that decomposes into training error, approximation error, and a diffusion-dependent term scaling as ${\log N}/{N}$, with an explicit rate obtained under a compositional drift assumption. Theoretical results are complemented by numerical experiments showing dimension-independent convergence in certain regimes and superior local-feature capture compared with a $B$-spline baseline, while also highlighting favorable memory and scalability properties of neural networks in higher dimensions. Overall, the method provides a scalable, theory-backed approach to drift estimation in high-dimensional diffusion models without requiring ergodicity, enabling effective use of discretely observed data from multiple independent trajectories.

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.

Figures (3)

• Figure 1: Convergence rates of our estimator for $d = 1$ (top left), $d = 2$ (top right), $d = 10$ (bottom left), and $d = 50$ (bottom right).
• Figure 2: Comparison of convergence rates for $d = 1$ (left) and $d = 2$ (right).
• Figure 3: Comparison of the ability to capture local fluctuations (N=5000). For $d = 2$, we display $\hat{f}$ (blue and orange) together with $f_0 - 5$ (green) to improve visual clarity.

Theorems & Definitions (18)

• Theorem 2.1
• Remark 2.2
• Corollary 2.3
• Proposition 4.1
• Lemma 4.2: Lemma A.2 in Denis2021
• Lemma 4.3: Lemma 4.13 in Oga2024
• proof : Proof of Proposition \ref{['prop:test-train-error-comparaison']}
• Proposition 4.4
• Lemma 4.5
• Lemma 4.6