# What are the optimal times to sample a process?

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.

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 $$t_n = T$$.
• Great answer! I think you meant $X/b=\frac{bY+Z}{1+b^2}$? Jan 30, 2022 at 11:57
• Huh, so it doesn’t depend on the earlier times.. did you mean $Z_t = bW_t - B_t$ instead though? I get that they are independent if $Z_t$ is defined to be that. Jan 30, 2022 at 12:50
• @NateRiver: I think it is correct with $Z_t=W_t-bB_t$: then $\mathbb EY_tZ_t=\mathbb E(bW_t^2+(1-b^2)B_tW_t-bB_t^2)=bt+0-bt=0$. But I just get my second coffee today, sorry if that makes no sense. Jan 30, 2022 at 13:24
• Oh, by the way: what is $\mu$ for? Jan 30, 2022 at 13:25