Variance of Stochastic Integral $\int_0^1{t^2 dWt}$

Question

I want to find Variance of integral $\int_0^1{t^2 dWt}$

$W_t$ is Brownian motion

What I did:

I used Ito formula and got:

$\int_0^1{t^2 dWt} = W_1 - \int_0^1 2tW_t dt$ (correct me if I'm wrong)

I do not know how to compute $\int_0^1 2tW_t dt$ or compute variance for whole answer.

Answer 1

Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,

$$ \mathbb{V}ar\left[\int_0^1 t^2 d W_t \right]= \mathbb{E}\left[\left(\int_0^1 t^2 d W_t\right)^2\right]$$

Applying Ito Isometry, we obtain

$$ \mathbb{E}\left[\left(\int_0^1 t^2 d W_t\right)^2\right] = \mathbb{E}\left[\int_0^1 t^4 dt\right] = \int_0^1 t^4 dt = \frac{1}{5} $$