Let $(\xi(t),t\ge 0)$ be an Ito process with $\xi(0)=\theta$ and $$ d\xi(t)=\kappa(\theta-\xi(t))dt+\sigma\sqrt{\xi(t)}dW(t)$$ for $\kappa, \theta, \sigma \in \mathbb{R}$. Show that $(\eta(t),t\ge 0)$, where $\eta(t):=t\xi(t)^2$, is an Ito process and give the stochastic differential $d\eta(t)$.
I know one can define $F(t,x):=t x^2$ and then apply the generalized Ito-formula to get $$d\eta(t)=\left(\xi(t)^2+2t\xi(t)\kappa(\theta-\xi(t))+t\sigma^2\xi(t)\right)dt+2t\sigma\xi(t)^{\frac{3}{2}}dW(t).$$
Since $\xi(t)$ is an Ito process it holds that $$\mathbb{E}\left[\int_0^T\sigma^2\xi(t)dt \right]<\infty \text{, for every } T>0.$$
However, in order to show that $\eta(t)$ is indeed an Ito process we need to show that $$\mathbb{E}\left[\int_0^T4t^2\sigma^2\xi(t)^3dt \right]<\infty \text{, for every } T>0.$$
This is where I am stuck. Is there something I am missing when $\sigma\ne 0$?
In the end I want to show that $$\int_0^T\mathbb{E}\left[\xi(t)^3\right]dt<\infty \text{, for every } T>0.$$