AI 中文总结
研究删失下多维随机微分方程的漂移估计问题，通过构建Nadaraya-Watson型漂移估计器及数据驱动带宽选择规则来进行估计，还推导了最优不等式和极小极大下界。
AI 中文摘要
在许多应用中，扩散轨迹仅在其保持在时变区域内时无失真地被观测到；在该区域外，仅记录其最近边界点和可见性指标。我们通过多维随机微分方程的删失方案来形式化此设置，其中通过将潜在状态欧几里得投影到随机、时变的闭凸集上来观测它。经典论证处理一维情况。然而，在高维中，投影会产生额外的有限变差项，并且在删失下漂移估计所需的局部信息是否保留并不明显。我们证明它是保留的。在投影映射的适当正则条件下，允许删失区域不光滑，观测过程是连续半鞅并允许广义伊藤分解，其有限变差校正项在可见时间上不分配质量。因此，可见时间上的随机积分对于观测和潜在过程恰好一致。基于此恒等式，我们构建具有数据驱动带宽选择规则的Nadaraya-Watson型漂移估计器并建立最优不等式。我们还推导了全观测和删失实验的各向异性极小极大下界。
英文摘要
In many applications, a diffusion trajectory is observed without distortion only while it remains inside a time-varying region; outside this region, only its nearest boundary point and an indicator of visibility are recorded. We formalize this setting through a censoring scheme for multidimensional stochastic differential equations, in which the latent state is observed via its Euclidean projection onto a random, time-dependent closed convex set. Classical arguments handle the one-dimensional case. In higher dimensions, however, the projection generates additional finite-variation terms, and it is not immediate that the local information required for drift estimation is preserved under censoring. We show that it is. Under a suitable regularity condition on the projection map, allowing the censoring region to be nonsmooth, the observed process is a continuous semimartingale and admits a generalized It{ô} decomposition, whose finite-variation correction term assigns no mass to visible times. Consequently, stochastic integrals over visible times coincide exactly for the observed and latent processes. Building on this identity, we construct a Nadaraya-Watson-type drift estimator with a data-driven bandwidth selection rule and establish oracle inequalities. We also derive anisotropic minimax lower bounds for both the full-observation and censored experiments.
