I've learned that for each continuous local martingale $M$, there's a unique continuous adapted non-decreasing process $[M]$ such that $M^2-[M]$ is a continuous local martingale.
For a local martingale $M$, is there a adapted non-decreasing process $[M]$ such that $M^2-[M]$ is a local martingale? (i.e. Do we have an analogous result for discontinuous local martingales?)
Thank you.
(The notes I have only consider the continuous case. I tried to adapt the argument, but ran into various problems...)