I'm new to stochastic calculus and I did an exercise but I don't know if it is correct, so I need somebody with more experience to check if it is true.
I am trying to compute the variance of the following random variable:
$$Z=\int _0^T e^{W_t} dW_t$$
So we have:
$\text{Var}(Z)=\text{Var}\left(\int _0^T e^{W_t} dW_t\right)$
By Itō's isometry we have:
$$\mathbb{E}\left[\int _0^T e^{2W_t} dt\right]$$
we can then bring inside the expectation to get:
$$\int _0^T \mathbb{E}\left[e^{2W_t}\right] dt = \int_0^T e^{2t} dt = \frac{e^{2T}}{2}-\frac{1}{2}$$
Moreover, if the above result is correct, what should I get instead of the problem asked me to compute
$$\text{Var}\left(\int _0^T e^{W_t} dt \right)$$
It should simply be the variance of a lognormal distributed random variable, computed in the extrema of the interval, or not?