Let $W=\{W_t\}_{t\geq 0}$ be a Brownian motion and $\{X_s\}_{t\leq s\leq1}$ be a Brownian bridge. Let we have a value function $V^*:[0,1)\times\mathbb{R}\cup\{(0,1)\}\rightarrow \mathbb{R}$ given by \begin{equation}
V^*(t,x)= \begin{cases}
\sqrt{2\pi(1-t)}(1-B^2)e^{x^2/2(1-t)}\Phi(\frac{x}{\sqrt{1-t}}), & \text{if } x<b(t) \\
x, & \text{if } x\geq b(t)
\end{cases},
\end{equation}
for $t<1$ and $V(1,0)=0$, where $b(t)=B\sqrt{1-t}$.
Lets show that $M_s=\int_{t}^{s}V_x^*(u,X_u)I(X_u\neq b(u))dW_u$ is a local martingal. In fact, since $V_x^*$ is bounded, the proces $M$ is a martingale.
Are there ''standard'' routes to follow when trying to prove that a process is/isn't a local martingale? How can I prove this for my case? Any help is appreciated!