发表机构
The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用稀疏ReLU神经网络估计多维Lévy驱动随机微分方程的漂移项，在弱集中条件下获得与扩散模型同阶的统计界，并证明估计器在极小极大意义下最优。
AI 中文摘要
我们针对离散观测的多维Lévy驱动的随机微分方程，使用稀疏ReLU神经网络，发展了非参数漂移估计的非渐近理论。在指数β混合和Lévy测度的指数尾部条件下，我们建立了最小二乘估计量的oracle不等式。跳跃分量相对于扩散模型改变了鞅结构和集中机制：连续鞅被不连续鞅取代，而相关的经验过程波动是亚指数而非亚高斯的。我们在不截断观测增量的情况下处理这些困难，使用补偿泊松积分的指数超鞅论证和ψ1链式方法。尽管集中性较弱，所得的统计界在常数范围内与相应的扩散界同阶。对于具有层次组合结构的漂移函数，这产生了内在维度的收敛速率。该框架允许无限跳跃活动，在某些情况下允许无限变差。最后，我们在一个固定的非平凡复合泊松子模型上建立了极小极大下界。结合上界，这表明估计器在对数因子范围内是极小极大最优的。
英文摘要
We develop a non-asymptotic theory for nonparametric drift estimation in discretely observed multi-dimensional Lévy-driven stochastic differential equations using sparse ReLU neural networks. Under exponential \(β\)-mixing and an exponential-tail condition on the Lévy measure, we establish an oracle inequality for the least-squares estimator. The jump component changes both the martingale structure and the concentration regime relative to diffusion models: continuous martingales are replaced by discontinuous martingales, while the relevant empirical-process fluctuations are sub-exponential rather than sub-Gaussian. We handle these difficulties without truncating the observed increments, using an exponential-supermartingale argument for compensated Poisson integrals and \(ψ_1\)-chaining. Despite the weaker concentration, the resulting statistical bound is, up to constants, of the same order as the corresponding diffusion bound. For drift functions with hierarchical compositional structure, this yields intrinsic-dimensional convergence rates. The framework allows infinite jump activity and, in some cases, infinite variation. Finally, we establish a minimax lower bound on a fixed non-trivial compound-Poisson submodel. Together with the upper bound, this shows that the estimator is minimax optimal up to logarithmic factors.
Comments58 pages
