I am quite curious to know if the following is true, which comes up to my mind when reading a paper on SLE:
For any local martingale $(X_t)_{t \geq 0}$ and stopping time $\tau$, is it true that $$ \big( X_t \mathbf{1}_ { \{t < \tau \} } + X_{\tau} \mathbf{1}_ { \{t \geq \tau \} } \big)_{t \geq 0}$$ is also a local martingale? The paper seems to use this fact implicitly.
But I don't really know how to handle the term $\mathbf{1}_ { \{{\tau_n} \wedge t < \tau \} }$, where $\{\tau_n \}$ is a reducing sequence. Any ideas?
Note that $X_{t \wedge \tau} 1_{t < \tau} = X_t 1_{t < \tau}$ and $X_{t \wedge \tau} 1_{t \geq \tau} = X_\tau 1_{t \geq \tau}$. Thus the sum you quote is simply an expansion of $X_{t \wedge \tau}$ which certainly is another local martingale.