I'm confused with different version os the Doob-Meyer decomposition. For example, in the book by Protter, p.116, Theorem 16 it is given for every cadlag supermartingale $Z=Z_0+M-A$ where $M$ is local martingale and $A$ is increasing predictable process.
The first question is whether $Z_0$ is a random variable, or deterministic value, i.e. whether the process $Z$ starts randomly.
The second question is why in many other books the Doob-Meyer decomposition is claimed to be for submartingales, e.g. given a submartingale $Z$, we have $Z=A+M$, where $A$ is increasing predictable and $M$ is martingale.