Some questions on Doob-Meyer decomposition

Question

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.

Answer 1

• $Z_0$ may be a random variable. It has to be measurable with respect to the $\sigma$-algebra $\mathcal{F}_0$ given by the corresponding filtration $(\mathcal{F}_t)_{t \geq 0}$.
• Suppose that you know the Doob-Meyer decomposition for supermartingales, i.e. for any supermartingale $Z$ there exists a local martingale $M$ and an increasing predictable process $A$ such that $$Z = Z_0+M-A.$$ Now let $Y$ be a submartingale. Then Doob-Meyer's decomposition applies to the supermartingale $Z:=-Y$, so there exist $M$ and $A$ as above such that $$-Y = -Y_0 + M-A,$$ i.e. $$Y = Y_0-M+A.$$ Obviously, $-M$ is a local martingale and therefore we have shown that $$Y= Y_0+N+A$$ for some local martingale $N$ and an increasing predictable process $A$. This shows that both formulations of the Doob-Meyer decomposition are equivalent.