Variance of stochastic integral of brownian motion

Question

How do i compute this integral?

$ Var [\int_0^T W(t)dW(t)] $

I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above

Answer 1

Let

$$X := \int_0^T W(t) \, dW(t)$$

By definition,

$$\text{var}X = \mathbb{E}((X-\mathbb{E}X)^2)$$

You already figured out that $\mathbb{E}X=0$, i.e.

$$\text{var} X = \mathbb{E}(X^2) = \mathbb{E} \left[ \left( \int_0^T W(t) \, dW(t) \right)^2 \right]$$

Hint Apply Itô's isometry.