What are the optimal times to sample a process?

Question

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.

Answer 1

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 $t_n = T$.