4th moment of stochastic integral?

Question

Hi I want to evaluate the expectation of 4th moment of the stochastic integral $$E \left[ \left( \int_0^T g(t)dW(t) \right)^4 \right]$$ and I found the following useful website 4th moment of a Wiener stochastic integral?. But I'm confused the computation of expectation of it: how can we deduce like this, $$ \mathbb{E}\left(X_t^4\right)=6 \int_0^t \int_0^s g(r)^2 g(s)^2 d r d s=3 \int_0^t \int_0^t g(r)^2 g(s)^2 d r d s. $$ The reason why I ask a new question instead of setting a comment is that I don't have enough reputation.

Any help will be appreciated!

Does this answer your question? 4th moment of a Wiener stochastic integral? — Kurt G., Nov 21 '22 at 12:37
Hello, I think no. I put the confused step of this answer in my question — Candlelight, Nov 21 '22 at 12:39
I have numbered the equations in the question you linked. What exactly is the confusion? — Kurt G., Nov 21 '22 at 12:41
Thank you for your help. I'm confused about the equation (5). I don't understand how we can change the upper limit of that integral $s$ to $t$ and the constant becomes $3$ from $6$. — Candlelight, Nov 21 '22 at 12:44
Hint: in the step from $6...$ to $3...$ try in integration-by-parts. It could work wonders. — Kurt G., Nov 21 '22 at 12:47
Thank you for your response. I'm confused about applying the integration-by-parts to double integral. And also if we apply it successfully, the upper limit$s$ will remain I think? — Candlelight, Nov 21 '22 at 12:58
Hint: set $u(s)=\int_0^sg^2(r),dr$ and $v(s)=g^2(s)$. Now integrate by parts. — Kurt G., Nov 21 '22 at 13:12
Since $u'(s)=v(s)$ you can also substitute to the new integration variable $u$. and realise that $\int_0^t\int_0^tg(r)^2g(s)^2,dr,ds$ is $u^2(t)$. — Kurt G., Nov 21 '22 at 13:14
Many thanks for the useful hint and sorry for the delayed response. I think I have figured it now. By the way, for equation (4), the author said we need to use Itô's formula. Can we just use Itô isometry to deduce it? — Candlelight, Nov 22 '22 at 11:36

4th moment of stochastic integral?

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