Let $W$ be a standard Brownian motion and $\mathcal F_t$ its natural filtration.

Suppose $\theta, A$ are positive random variables independent of $\mathcal F_t$. Let $Y_t$ be the process

$$Y_t := A \sin \, (\theta t) + W_t $$

and denote by $\mathcal Y_t$ its natural filtration.

Question: Is it true that $\mathbb E[A| \mathcal Y_t] \to A$, and $\mathbb E[\theta| \mathcal Y_t] \to \theta$ almost surely?

Let $W$ be a standard Brownian motion and $\mathcal F_t$ its natural filtration.

Suppose $\theta, A$ are positive $L^1$ random variables independent of $\mathcal F_t$. 

Let $Y_t$ be the process

$$Y_t := A \sin \, (\theta t) + W_t $$

and denote by $\mathcal Y_t$ its natural filtration.

Question: Is it true that $\mathbb E[A| \mathcal Y_t] \to A$, and $\mathbb E[\theta| \mathcal Y_t] \to \theta$ almost surely?