This seems counter-intuitive to me since variance is a difference of expectations and afaik, unconditional expectation is a real number.
Apparently, $X_t$ where $dX_t = Y_t dW_t$, where $Y_t$ is an independent Brownian motion to $W_t$, has random variance.
Solving the SDE gives $X_t = X_0 + \int_0^t Y_s dW_s$
Computing the first and second moments give:
$E[X_t] = E[X_0] + E[\int_0^t Y_s dW_s]$
$E[X_t^2] = E[X_0^2] + 2E[X_0 \int_0^t Y_s dW_s] + E[\int_0^t Y_s^2 ds]$ by Itō isometry
$ = E[X_0^2] + 2E[X_0 \int_0^t Y_s dW_s] + \int_0^t E[Y_s^2] ds$ by Tonelli's theorem
I'm guessing that $E[\int_0^t Y_s dW_s]$ or $E[X_0 \int_0^t Y_s dW_s]$ is random? Why? It seems that $\int_0^t Y_s dW_s$ is a random variable with a unique mean given by $E[\int_0^t Y_s dW_s]$.
What am I not getting here?