For an Ornstein-Uhlenbeck process of the form
\begin{equation} dx=a(t)x(t)dt+N(t)dW, \end{equation}
Suppose $y(t)=x(t+\Delta t)$. I am trying to calculate $\mathbb{E}[xy]$ using Ito's lemma: $$d[xy]=dx*y+x*dy+dxdy.$$ So far, I have $$d[xy]=a(t)xydt+N(t)ydW(t)+a(t+\Delta t) xy dt+N(t+\Delta t) x dW(t+\Delta t)+N(t)N(t+\Delta t) dt.$$
I am not sure how to proceed. I think that $\mathbb{E}[x(t) dW(t)]=0$, but what is $\mathbb{E}[x(t) dW(t+\Delta t)]$? What about $\mathbb{E} [x(t+\Delta t)dW(t)]$?
Thanks in advance!