I suppose that I got the correct solution, but I would like you to confirm me.
I started with an explicit solution of the SDE
because this integral is an Ito integral and it is martingale
So the mean is:
and for the variance I got:
I used that
It depends on what is . If it is constant, then you are correct (and ). If it is a random variable, then it must appear in the variance: and both terms are independent and centered hence = . I think.