Question about a step in the proof of the bracket process

Question

Let's first state the theorem

$\forall M$ continuous local martingale, there exists a unique increasing continuous process $\langle M\rangle $ zero at $t=0$ and such that $M^2-\langle M \rangle $ is again a continuous local martingale. Further for all stopping times $\tau$, we have $\langle M^\tau\rangle = \langle M \rangle^\tau$.

The last statement is not clear for me. I know that $L:=M^2-\langle M \rangle $ is a local martingale. Then we look at $L^\tau=(M^\tau)^2-\langle M \rangle^\tau $, and we apply the stopping theorem. Why do we can apply the stopping theorem? For that, $L$ should be uniformly integrable, or $\tau$ must be finite?

thanks for your help

Answer 1

All you need to use is that the set of local martingales are stable under stopping, i.e. $M^\tau$ is local martingale if $M$ is a local martingale and $\tau$ is any stopping time. Then $$ (M^{\tau})^2-\langle M\rangle^\tau=(M^2-\langle M\rangle)^{\tau} $$ yields the result.