I applied Itô's formula to $f(x)=x^q$ for $q>1$ with a process defined by the SDE \begin{align*} dX_t = \mu(t,X_t)dt + \sigma(t,X_t) dW_s, \end{align*} where $W_s$ is a Brownian motion, $\mu(t,X_t)$ is the drift, which obeys the growth condition $\mu(t,x)\leq C (1+|x|)$, for some constant C>0, and $|\sigma|$ is bounded.
Now, assuming that we look at $X_t$ up to the first hitting time of $X_t=0$, $f\in\mathcal{C}^2(\mathbb{R}_+)$ so Itô can be applied, i.e. \begin{align*} \mathbb{E}[X_t^q]=\mathbb{E}[\int_0^t qX_s^{q-1}dX_s]+\mathbb{E}[\frac{1}{2}\int_0^t(q-1)qX_s^{q-2}d[X]_s], \end{align*}
where $[X]_s$ is the sharp bracket of $X_t$. I'd like to deduce that \begin{align*} \mathbb{E}[X_t^q]=c\cdot\mathbb{E}[\int_0^t X_s^qds], \end{align*} for some constant $c>0$.