I have a questions about Exercise3.3.25 at Karatzas&Shreve"Brownian Motion and Stochastic Integral".
$W=\{W_t, F_t; 0\leq t<\infty\}$ is a standard, one-dimensional Brownian motion and X a measurable, adapted process satisfying \begin{equation} E\int_{0}^{T}|X_t|^{2m}dt < \infty \end{equation} for some real numbers $T > 0$ and $m \geq 1$, show that \begin{equation} E\left|\int_{0}^{T}X_t dW_t\right|^{2m}\leq (m(2m-1))^mT^{m-1}E\int_{0}^{T}|X_t|^{2m}dt \end{equation}
Hint: $ \{M_t=\int_{0}^{t} X_sdW_s, F_t; 0\leq t \leq \infty \} $ is martingale, and use Ito's rule for $|M_t|^{2m}$
Why we can use absolute function for Ito's rule. I think we cannot use it directly bacause abosolute function is not differentiable at 0.