Abstract: By Doob-Meyer decomposition theorem, for any càdlàg supermartingale $Z$, there exists a unique predictable increasing process $A$ starts from $A_0=0$ such that $Z+A$ is a local martingale starts from $Z_0$. (for example, Protter (2005) Theorem 16 page 116, Dellacherie and Meyer (1982) Theorem 12 page 198).
If removing the condition for $A$ to be predictable, can we always find a martingale $M'$ (not a local martingale) such that $Z=Z_0+M'-A$?
Backgrounds: In the Corollary 4.5 (page 19) of the lecture note by Martin Hairer, the following claim is presented:
Let $\{X_t\}$ be a continuous-time Markov process with càdlàg paths on a Polish space $E$ and let $F,G:[0,\infty)\times E\to\mathbb{R}$ be continuous functions such that $$ F(t,X_t)-\int_0^tG(s,X_s)\,ds $$ is a suparmartingale for any starting point $x_0\in E$. Then, if $\Phi\in C^{1,2}([0,\infty)\times\mathbb{R})$ is $C^1$ in its first argument and $C^2$, concave, and increasing in its second argument, we have that $$ \Phi(t,F(t,X_t))-\int_0^t\biggl(\partial_t\Phi(s,F(s,X_s))+\partial_x\Phi(F(s,X_s))G(s,X_s)\biggr)ds $$ is a suparmartingale.
The proof starts by claiming that, setting $Y_t:=F(t,X_t)$, there are a càdlàg martingale $M$ and a non-increasing process $N$ such that $$ dY_t=G(t,X_t)dt+dN_t+dM_t. $$ However, this requires in general that for any càdlàg supermartingale $Z_t:=Y_t-\int^t_0G(s,X_s)ds$, there is a decomposition $Z_t=Z_0+N_t+M_t$ into càdlàg martingale $M$ and non-increasing $N$.
Am I missing any hidden assumption (such as $\{Z_t\}$ is of class (D))? Or, the general assertion is true?
Any help is appreciated. Thanks in advance.
