Sup-norm adaptive drift estimation for multivariate nonreversible diffusions
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多元非可逆扩散的上确界范数自适应漂移估计
- 研究问题
- 对连续观测的遍历扩散过程，如何在未知平滑度和可能非可逆的条件下，以最大绝对误差衡量并自适应估计漂移？
- 方法与思路
- 作者利用混合性质替代可逆性相关工具构造自适应估计，分别处理有界域上的多元扩散和实线上可含无界漂移的标量扩散。
- 材料陈述的贡献
- 摘要称获得最优上确界收敛速率、标量情形的极小极大结果及下界，并建立标量不变密度核估计的 Donsker 定理与效率结论。
- 阅读关注点
- 应区分标量与多元结果，核对各自的混合和平滑条件、估计区域、自适应规则，以及最优速率与下界是否对应同一模型类。
- 阅读边界
- 所据 OpenAlex 索引摘要未注明稿件版本；各定理具体条件和速率未列出，不能把标量全实线结论推广到多元情形。
原文摘要
We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup-norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup-norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisfied for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β-mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efficiency.