I currently read paper [2], where, in Chapter 2, one shows that $Y_t-y_t(\vartheta_0)=\varepsilon \xi_t$, where $\xi_t$ is a Gaussian process with $\mathbb{E}_{\vartheta_0}\xi_t=0$. Now, they claim that for every $p>0$ it holds that $$\mathbb{E}_{\vartheta_0}|\xi_t|^p<Ct^{p/2}.$$
Why does this inequality hold? The author names [1] as reference and the only Theorem in that book going into that direction is Lemma 4.11 which states that $\mathbb{E}(\int_0^t f(s,\omega)dW_s)^{2m}\leq K^{2m}t^m(2m-1)!!$ for $W$ a Wiener process and $|f(t,\omega)|<K$ a bounded function. But in my opinion this result is not applicable here. Do you have any other idea on how to show this result?
References:
[1] Statistics of Random Processes, Liptser & Shiryaev, 2nd ed.
[2] On parameter estimation of the hidden Gaussian process in perturbed SDE, Kutoyants, Zhou, https://arxiv.org/pdf/1904.09750.pdf