Distribution of stochastic integral

Question

I am trying to find the distribution of the stochastic integral $X_t= \int^1_0\sqrt2 W_t dW_t $ where $W_t$ is a brownian motion.

I know this will be normal, so I have started to compute the moments for this: $E(X_t)= 0$ since $W_t $~$ N(0,t)$ But I am stuck on the variance. $$Var(X_t)=E(X_t^2) \\ =2\iint^1_0E(W_tW_s)dW_sdW_t$$ I know $E(W_tW_s)=min(t,s)$, but how do I compute the integral?

Am I on the right lines?

Answer 1

Using the Ito isometry and Fubini's theorem, we have $$ \mathbb{E}\Big[\Big(\int_0^1\sqrt{2}W_t\;dW_t\Big)^2\Big]=\mathbb{E}\Big[\int_0^1(\sqrt{2}W_t)^2\;dt\Big]=\int_0^12\mathbb{E}[W_t^2]\;dt=\int_0^12t\;dt=1 $$

Another approach is to use Ito's lemma to obtain $$ \int_0^1\sqrt{2}W_t\;dW_t=\frac{\sqrt{2}}{2}(W_1^2-1)$$ and then compute $\frac{1}{2}\mathbb{E}[(W_1^2-1)^2]$. By the way, this calculation shows that $X_t$ is not normally distributed.