Question

Doob’s Decomposition: Let $(X_m)_{m \geq o}$ be a submartingale relative to the filtration $F_0 \subset F1 \subset \dotsm$ .

Show that there is a martingale $(M_m)_{m \geq 0}$   and a predictable sequence $(A_m)_{m \geq 0}$ such that $X_m =M_m+A_m$ for all $m \geq 0$ and $0=A_0 \leq A_ 1 \leq A_2 \leq \dotsm$ .

Answer 1