What is the mean and the variance of $y_t$, given the following SDE:
$dy_t = -x_t y_t dt + \sigma_1 dW^1_t$
$dx_t = -\sigma_2 y_t dW^2_t$
$W^1$ and $W^2$ are (possibly correlated) Wiener processes.
Especially, is it true that $Var[dy_t] = \sigma_1^2 dt$, or is that not necessarily correct due to the process $x_t$ in the drift of $y_t$?