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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

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see example 1 at in this case $X_t = -1 , t > 1$ – mike Jul 18 '12 at 23:14
This question seems like it is asking 'Is a local martingale a martingale?' – jwg Mar 4 at 15:48

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