Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions

研究问题: 如何从多维扩散过程的连续观测轨迹估计周期性 Lipschitz 漂移向量场，并刻画后验分布与不变测度估计的渐近性质？
方法与思路: 作者以高维高斯乘积先验构造后验分布，由后验收缩推导惩罚最小二乘暨 MAP 估计的收敛率，并研究低维情形的后验极限定理。
材料陈述的贡献: 摘要称，平方损失下各维度、上确界损失下至多四维的速率达到对数因子内最优，并在至多三维建立非参数 Bernstein–von Mises 定理。
阅读关注点: 应核对先验与漂移平滑性的具体条件、各损失下的维度限制，以及后验极限定理如何导出不变测度估计的函数型中心极限定理。
阅读边界: 仅据版本未注明的 OpenAlex 索引摘要；先验尺度、函数空间和正则条件未说明，不能将低维极限定理直接推广到更高维。

GPT-6.1 Sol · 高 · 2026-10-07

原文摘要

The problem of determining a periodic Lipschitz vector field $b=(b_{1},\ldots ,b_{d})$ from an observed trajectory of the solution $(X_{t}:0\le t\le T)$ of the multi-dimensional stochastic differential equation $\begin{equation*}dX_{t}=b(X_{t})\,dt+dW_{t},\quad t\geq 0,\end{equation*}$ where $W_{t}$ is a standard $d$ -dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in $L^{2}$ -loss in any dimension, and also for supremum norm loss when $d\le 4$ . Further, when $d\le 3$ , nonparametric Bernstein–von Mises theorems are proved for the posterior distributions of $b$ . From this, we deduce functional central limit theorems for the implied estimators of the invariant measure $\mu _{b}$ . The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.The problem of determining a periodic Lipschitz vector field $b=(b_{1},\ldots ,b_{d})$ from an observed trajectory of the solution $(X_{t}:0\le t\le T)$ of the multi-dimensional stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dW_{t},\quad t\geq 0,\end{equation*} where $W_{t}$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in $L^{2}$-loss in any dimension, and also for supremum norm loss when $d\le 4$. Further, when $d\le 3$, nonparametric Bernstein–von Mises theorems are proved for the posterior distributions of $b$. From this, we deduce functional central limit theorems for the implied estimators of the invariant measure $\mu _{b}$. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）

笔记与分组

分类依据与纠错

Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions

代码、数据与补充材料

多维扩散过程漂移向量场的非参数统计推断

原文摘要

同方向的基础文献

同方向的基础文献