Need continuous time martingales be predictable?

Question

I know that for discrete time martingales, one can show that if they are predictable, they have to be constant. Now for continuous time martingales, that is not true I suppose, since for example a Brownian motion is predictable as it is continuous (correct?). So I was wondering what the relationship between continuous time martingales and predictability is. I could not come up with an example of a non-predictable martingale. Are all continuous time martingales predictable? If not, what would be a counterexample?

Answer 1

The following may be near what you are after: If $M$ is a (cadlag) continuous time martingale, then ${}^pM$ coincides with $M_-$ almost surely. Here ${}^pM$, the predictable projection of $M$, is characterized as the predictable process such that $E[M_T\mid\mathcal F_{T-}] = {}^pM_{T}$ for all bounded predictable stopping times $T$; and $M_-$ is the left limit process $t\mapsto M_{t-}$. In particular, this shows that $M$ is predictable if and only if the paths of $M$ are continuous.