How can I calculate the expected value $\mathbb{E}(I_{110}^2 * I_{10})$, where $I_{110}$=$\int_{t_0}^T \int_{t_0}^{s_3} \int_{t_0}^{s_2} 1\, \, dW(s_1) dW(s_2) ds_3$ and $I_{10}=\int_{t_0}^T \int_{t_0}^{s_2} 1 \, \, dW(s_1) ds_2$ are Ito Integrals, W is a 1-dimensional Wiener Process and the intevrall is $h:=[t_0,T]$?
Thank you.
Define a second Brownian motion $V$ by $V_t=W_t$ for $t\leqslant t_0$ and $V_t=2W_{t_0}-W_t$ for $t\geqslant t_0$ and consider the integrals $J_{110}$ and $J_{10}$ corresponding to $I_{110}$ and $I_{10}$ using $V$ instead of $W$.
Then $J_{110}=I_{110}$ and $J_{10}=-I_{10}$ hence $J_{10}J_{110}^2=-I_{10}I_{110}^2$. Since $J_{10}J_{110}^2$ and $I_{10}I_{110}^2$ have the same distribution, the distribution of $I_{10}I_{110}^2$ is symmetric, in particular $E[I_{10}I_{110}^2]=0$.