Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\ X_0=x_0 \\ dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^2 \\ Y_0=y_0 \end{cases} \end{equation} where $W^1,W^2$ are Brownian motions.
Now take a time $s>0$, since $X_s,Y_s$ are random variables it should make sense to compute $\mathbb{E}[X_{s}|Y_s ]$.
My question are:
can I apply the tower property in this way? $$\mathbb{E}[X_{s}|Y_s ]=\mathbb{E}[\mathbb{E}[X_{s}|(Y_\tau, \tau \leq s) ]|Y_s ]$$
What does it mean to compute $\mathbb{E}[X_{s}|Y_s ]$ when the process $X_s$ depends (by the sde) on the all history of $Y_\tau, \tau \leq s$?
