Define a process $X = (X_{t})_{t\ge0}$ by $X_{t} = W_{t} - t)exp(W_{t} - t/2)$. How do you find $E[X_{t}]$ for $t\ge0$ using Ito's lemma?
I tried $X_{t} - X_{0} = \frac{1}{2}\int_{0}^{s} X''_{t}dt + \int_{0}^{s} X'_{t}dW_{t}$ and got $E[X_{t}] = \frac{1}{2}E[\int_{0}^{s}[exp(W_{t} -t/2) + \frac{1}{4}(W_{t}-t)exp(W_{t}-t/2)]]$
But I don't know where to proceed from here - did I use Ito's Lemma wrong?