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I am eager to compute the following variance, where $W_t$ is a Wiener process:

$$\mathbb{V}\left[I_t\right], \text{ where } I_t = \int\limits_0^tW_u^2 u^2\partial u.$$

I have already computed $\mathbb{E}[I_t] = \Large\frac{t^4}{4}$, and hence currently I have:

$$ \mathbb{V}[I_t] = \mathbb{E}[I^2_t] - \mathbb{E}^2[I_t] = \mathbb{E}[I^2_t] - \frac{t^{16}}{16} = \mathbb{E}\left[\left(\int\limits_0^tW_u^2 u^2\partial u\right)^2\right] - \frac{t^{8}}{16}. $$

And here, I do not understand, how to compute the first summand. I thought of applying the Itô isometry here, but as long as $W_u^2u^2$ is a stochastic process (at least it seems for me it is, what may be verified), the differential is taken not over a Wiener process, but rather over the time-like variable $u$, so that I have no further ideas on how to proceed.

Any help would be appreciated, thank you in advance!

Ѕᴀᴀᴅ
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o.spectrum
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  • A generic strategy is integration by parts. Here, you can replace $u^2$ with $2u \partial u$ and $W_t^2$ with $\int W_t^2 \partial u$, which is discussed in several places on this side, including this one. – Eric Towers Dec 04 '21 at 14:12
  • @EricTowers Thank you for sharing such an idea, but what if, for example the integral itself was way harder? Is there any other approach to get rid of the squared term without involving the integral computation itself? Just I want to understand, what could the possible strategy be, if I am "unable" or "forbidden" to compute $I_t$. – o.spectrum Dec 04 '21 at 14:15

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I have limited time just now, and can fill in more details later if you want. Idea comes from here $$\Bbb E[( \int_0^t W_u^2 u^2 d u)^2 ] = \Bbb E[ \int_0^t W_u^2 u^2 d u \cdot \int_0^t W_s^2 s^2 d s ] = \Bbb E [\int_0^t\int_0^t W_s^2 W_u^2 s^2u^2 d u ds ]\\ =\int_0^t\int_0^t \Bbb E[ W_s^2 W_u^2 ] s^2u^2 d u ds $$ and

$$E[ W_s^2 W_u^2 ] = E[ W_{\min (s,u)}^2 (W_{\max(s,u)}- W_{\min (s,u)} + W_{\min (s,u)})^2 ] \\ = \Bbb E[W_{\min (s,u)}^4] + \Bbb E[W_{\min (s,u)}^2]\Bbb E[(W_{\max(s,u)} -W_{\min (s,u)})^2], $$ which can be further and explicitly computed.

Falrach
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  • Wow, that is very explicit explanation, moreover once you have a limited time, thank you a lot! This idea seems very useful even for more complex integrands, thank you once again! – o.spectrum Dec 04 '21 at 14:45