The lower-bound conjecture for transformed SDE approximation under (A1)–(A3)

Let $x_0\in\mathbb{R}$, let $\mu,\sigma\colon\mathbb{R}\to\mathbb{R}$ satisfy (A1)–(A3), and let $X$ be the solution of the corresponding SDE. For $n\in\mathbb{N}$, observations consist of $W_{t_1},\dots,W_{t_n}$ at arbitrary times $t_1,\dots,t_n\in[0,1]$, followed by a measurable approximation $\psi\colon\mathbb{R}^n\to\mathbb{R}$. Lower-bound conjecture. There exist $x_0\in\mathbb{R}$, functions $\mu,\sigma\colon\mathbb{R}\to\mathbb{R}$ satisfying (A1)–(A3), and $c\in(0,\infty)$ such that, for every $n\in\mathbb{N}$,

$$\inf_{\substack{t_1,\dots,t_n\in[0,1]\\ \psi\colon\mathbb{R}^n\to\mathbb{R}\text{ measurable}}} \mathbb{E}\bigl[|X_1-\psi(W_{t_1},\dots,W_{t_n})|\bigr]\geq\frac{c}{n^{3/4}}.$$

This is proposed as a matching lower bound for the order $n^{-3/4}$ error estimate, showing that the rate cannot generally be improved by any method based on $n$ evaluations of the driving Brownian motion.