I am trying to apply Ito's lemma to compute variance of the following integral
$X(t) = \int_{0}^t W(s)dW(s),$
where $W(t)$ is a Wienner process. Could you please check my calculations?
$$E(X(t)) = 0 \\ E(X^2(t)) = E\left(\left( \int_{0}^t W(s)dW(s) \right)^2\right) = E\left( \int_{0}^t W(s)^2 ds \right) = \int_{0}^t E(W(s)^2) ds = \frac{t^2}{2}.$$
Hence, $Var(X(t)) = \frac{t^2}{2} - 0^2 = \frac{t^2}{2}.$
Thank you in advance.