Drift Estimation for Diffusion Processes Using Neural Networks Based on Discretely Observed Independent Paths
This work addresses drift estimation for diffusion processes, which is incremental as it applies neural networks to a known statistical problem with specific improvements in convergence and dimensionality handling.
The paper tackles nonparametric drift estimation for diffusion processes using neural networks based on discrete observations from independent paths, deriving a non-asymptotic convergence rate and showing in experiments that the method outperforms B-spline estimators, especially in higher dimensions.
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.