Compare Doob décomposition for stochastic process $(X_t)_{t\geq 0}$ and $(X_n)_{n\in\mathbb N}$

Question

If know that if $(X_n)_{n\in\mathbb N}$ is a stochastic process, then $X_n$ can be written as $X_n=M_n+A_n$ where $(M_n)$ is a Martingale and $(A_n)$ is an adapted and predictable process.

I know that for $(X_t)_{t\geq 0}$ being a semi-martingale, $$X_t=M_t+A_t$$ where $(M_t)$ is a local martingale and $A_t$ is an adapted and predictable process.

Question

So in discret version, such a decomposition exist for any process, whereas the theorem I have in continuous version is only for semimartingale. But at the end, does such decomposition works for any process as well like in the discrete version, or $(X_t)$ must be a semimartingale ?

Answer 1

The semi-martingale part of it is important. Recall that there's a tight relationship between local martingales and semi martingales (every local submartingale is a semimartingale). One verion of the theorem you reference is the Doob-Meyer decomposition that states that $X$ must be a class $(DL)$ right-continuous submartingale in order for a right-continuous martingale $M$ and a right-continuous increasing predictable process $A$ to exist, such that $X_t=M_t+A_t$. In this framework, $X$ being of $(DL)$ class requires that $\forall \alpha\geq 0$, denoting $\tau$ a given stopping time, $\{X_\tau: \tau\leq \alpha\}$ is uniformly integrable. The importance of this local property has to do with the usual technique in continuous time of proving results for local martingales on stopping times and then extending the result to any time (here, the uniform integrability plays a role in granting that the expectations are finite while extending through stopping times).

In your case, by not working with semi-martingales, your process might not have the local characteristics needed to make the Doob-Mayer decomposition be generally true for all times.

Here, you can find more information on $(DL)$ submartingales and the Doob-Meyer Decomposition.

Answer 2

It seems that two distinct notions are being conflated into one in your question.

1. A non-unique semimartingale decomposition, i.e., decomposition into a local martingale and a finite variation process;
2. The unique canonical decomposition of a special semimartingale, i.e., decomposition into a local martingale and a predictable finite variation process.

The first is available for all semimartingales and in discrete time we can just write $X = 0 + X$ because $X$ will always be of finite variation. The second decomposition is only available for a subclass of semimartingales (precisely the special semimartingales). So one way to interpret the Doob decomposition is to say, among other things, that a $(\mathrm{DL})$ submartingale is a special semimartingale.