Consider $X =(X_t)_{t \in [0T]}$ progressively measurable with $X_t \in \mathbb L^p, \forall t \in [0,T]$ for $p\geq 1$ and $f$ a $k$-Lipschitz function.
I would like to show that $(\int_0^t f(X_s) dW_s)_{t \in [0T]}$ is a continuous martingale , where $W$ is the brownian motion.
Indeed, since $f$ is a $k$-Lipschitz function and by Jensen's inequality
\begin{align} \int_0^T \left| f(X_s)\right|^2 ~ ds &\leq \int_0^T\left( \left| f(X_0)\right| + k \left| X_0\right|+ k \left|X_s\right|\right)^2 ~ds \\ & \leq 3 \int_0^T \left| f(X_0)\right|^2 + k \left| X_0\right|^2+ k \left|X_s\right|^2 ~ds \\ & = 6k^2 \left| X_0\right|^2T+ 3k^2 \int_0^T \left|X_s\right|^2 ~ds \\ \end{align} \begin{align} \mathbb E\left[\int_0^T \left| f(X_s)\right|^2 ~ ds\right]&\leq C_T\left( 1+ \mathbb E\left[\int_0^T \left|X_s\right|^2 ~ds\right]\right) \\& \leq C_T\left( 1+ T\ \mathbb E\left[\sup_{t\in [0,T]} \left|X_s\right|^2 \right]\right)< \infty \end{align}
with $C_T= \max \{6k^2 \mathbb E\left[ \left| X_0\right|^2\right]T ,3k \}$.
Could someone check if I made any mistake please ?