Simple Question:
If a process $X$ is a local martingale, can I immediately use the martingale property of
$$\mathbb{E}\left(X(t)|\mathcal{F}_s\right) = X(s)$$
because a local martingale is a driftless process? In the text I have, the Kazamaki or Novikov criterion is first proved before the above property is used, but is it necessary? If so, why?
Thanks for any help provided
