Let $X$ be a one dimensional Ito diffusion given by

$$X_t = b \,W_t$$

where $b$ is a constant, and $W$ is a standard Brownian motion.

Let $B$ be another Brownian motion independent of $W$, and define the *observation process* $Y$ by

$$Y_t = X_t + B_t.$$

Fix $T > 0$. A choice of *sampling times* is simply a choice of real numbers $0 \leq t_1 \leq \dots \leq t_n \leq T$.

**Question:** For fixed $n > 1$, what choice of sampling times $t_1, \dots, t_n$ minimises the expression

$$\mathbb E\left [ |\mathbb E[X_T| \sigma(Y_{t_1}, \dots, Y_{t_n})] - X_T | \right ]?$$

Where $\sigma(Y_{t_1}, \dots, Y_{t_n})$ denotes the sigma algebra generated by the $Y_{t_i}$.

**Remark:** It is not certain that there exists a minimiser - to prove existence first, it would suffice to show that the given function is continuous in the $t_i$ and apply compactness.