 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.