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.



What are the optimal times to sample a process?
Write $$Z\_t = W\_t - b B\_t,$$ so that $Y\_t$ and $Z\_t$ are independent Brownian motions, $$X\_t = b W\_t = b \cdot \frac{b Y\_t + Z\_t}{1 + b^2} \, ,$$ and the question asks for the distance between $X\_T$ and
$$ \mathbb E[X\_T | \sigma(Y\_{t\_1},\ldots,Y\_{t\_n})] = b \cdot \frac{b Y\_{t\_n} + 0}{1 + b^2} \, . $$
This distance is of course
$$ \frac{b}{1 + b^2} \mathbb E[|b (Y\_{t\_n} - Y\_T) - Z\_T|] , $$
which is minimised when $t\_n = T$.


