Context: In some lecture slides for a class I am taking, we are given the equations
$$d\langle x \rangle = \langle dx \rangle$$ $$d\langle x^2 \rangle = \langle 2\,x\,dx \rangle + \langle \frac{1}{2}\,(dx)^2 \rangle$$
and I am wondering the justifications for these equations.
More generally, consider and SDE of the form
$$ dx = a\left(t,\ x\right)\,dt + \sigma\left(t,\ x\right)\,dW $$
Is there a closed-form equation for $d\langle f\left(t,\ x\right)\rangle$ for some general $f(t,\ x)$? Possibly in terms of $\langle df\left(t,\ x\right) \rangle$?
My initial thought is to use Ito's formula
$$d f\left(t,\ x\right) = \left(\partial_t f + a(t,\ x)\,\partial_x f + \frac{1}{2}b(t,\ x)\,\partial_x^2 f\right)\,dt + b(t,\ x)\,\partial_x f\,dW$$
which leads to an equation for $\langle d f(t,\ x) \rangle$. However, I don't know how the differential operator $d$ interacts with the ensemble average $\langle \cdot \rangle$.