用神经网络估计扩散过程的漂移函数，高维下表现更优。
Drift Estimation for Diffusion Processes Using Neural Networks Based on Discretely Observed Independent Paths
- 基于独立轨迹的离散观测，设计神经网络漂移估计器。
- 收敛速率分解为训练、近似与扩散项，高维下可忽略维度影响。
- 适合高维复杂漂移结构建模，优于B样条方法。
本文研究时齐扩散过程在紧凑域上漂移函数的非参数估计问题，基于来自N个独立轨迹的高频离散观测。提出一种基于神经网络的估计方法，并推导出非渐近收敛速率，其由训练误差、近似误差和尺度为$\log N/N$的扩散相关项构成。对于组合型漂移函数，给出显式收敛率。数值实验中，采用双层组合结构生成具有局部振荡的漂移函数，结果显示经验收敛率与输入维度d无关。相比B样条方法，神经网络估计器在更高维场景下收敛更快，且更有效捕捉局部特征。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。