Doob decomposition theorem

Every integrable discrete-time supermartingale has the form $V=M-A$, where $M$ is a martingale and $A$ is predictable, integrable, nondecreasing, and starts at zero.

More generally, an integrable adapted process $X_n$ decomposes as $X_n=M_n+A_n$ with $A_0=0$ and $A_n-A_{n-1}=\mathbb{E}[X_n-X_{n-1}\mid {\mathcal{F}}_{n-1}]$. For a supermartingale this drift is nonincreasing, so writing it as $-A$ yields the nondecreasing convention above.

If a submartingale has uniformly bounded increments, then ${\mathrm{liminf}}_n{Y}_n/n\ge 0$ almost surely. In its Doob decomposition in discrete time, the predictable compensator is nondecreasing, while the martingale part has bounded increments. The strong law for martingales with bounded increments makes the latter negligible on the linear scale.

Every adapted integrable process $X$ has a unique decomposition $X=M+A$, where $M$ is a martingale, $A_0=0$, and $A$ is predictable. Its increments areFor a supermartingale these increments are nonpositive; changing the sign of $A$ gives the usual nondecreasing compensator convention.