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Distribution of standard Ito integral is well known: $$I_1(f) = \int_0^T f(t)dB(t) \sim \mathcal{N}\bigg(0, \int_0^T f^2(t)dt\bigg).$$ Is it possible to find the distribution of multiple Wiener-Ito integral: $$I_n(f) = \int_{[0,T]^n} f(t_1, \dots, t_n) dB(t_1) \cdots dB(t_n)$$

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    $\begingroup$ I don't know of a closed form expression, but there are certainly lots of results about such distributions. You might like to look up "Wiener chaos". $\endgroup$ – Nate Eldredge Aug 26 '18 at 19:22
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Below some references regarding distributional properties of Wiener chaoses

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