AI 中文总结
本文研究含噪信息下标量SDE的强逼近问题，提出基于含噪漂移、扩散及Wiener路径信息的随机Milstein格式，证明其$L^r$误差界及匹配极小极大下界，验证该格式为极小极大阶最优。
AI 中文摘要
我们研究了当漂移系数、扩散系数、扩散系数导数的可用标准信息以及观测到的Wiener路径受噪声干扰时，标量随机微分方程的强逼近问题。漂移信息、扩散信息和Wiener路径观测的精度由三个非负参数$δ_1,δ_2,δ_3$描述，其中$δ_2$同时控制含噪扩散系数以及Milstein校正所需的独立含噪导数预言机。我们分析了一种仅基于该噪声信息的随机Milstein格式，并证明当$r\geq 2$时，其$L^r$误差有界为$C(n^{-\min\{γ_1+1/2,γ_2\}}+δ_1+δ_2+δ_3)$，其中$n$为时间步数，$γ_1,γ_2$为系数的时间Hölder指数。我们还在本文所考虑的随机标准信息模型中证明了匹配的极小极大下界。特别地，与$δ_3$成正比的Wiener路径贡献是不可避免的，且该含噪随机Milstein格式是极小极大阶最优的。
英文摘要
We investigate the strong approximation of scalar stochastic differential equations when the available standard information about the drift coefficient, the diffusion coefficient, the derivative of the diffusion coefficient, and the observed Wiener path is corrupted by noise. The precision of the drift, diffusion-information, and Wiener-path observations is described by three nonnegative parameters $δ_1,δ_2,δ_3$, where $δ_2$ controls both the noisy diffusion coefficient and the separate noisy derivative oracle required in the Milstein correction. We analyze a randomized Milstein scheme based only on this noisy information and prove, for $r\geq 2$, that its $L^r$-error is bounded by $C(n^{-\min\{γ_1+1/2,γ_2\}}+δ_1+δ_2+δ_3)$, where $n$ is the number of time steps and $γ_1,γ_2$ are the temporal Hölder exponents of the coefficients. We also prove a matching minimax lower bound in the randomized standard-information model considered in the paper. In particular, the Wiener-path contribution proportional to $δ_3$ is unavoidable, and the noisy randomized Milstein scheme is minimax order-optimal.
Comments33 pages, 5 figures, 2 tables
