Doob decomposition theorem

Remarks

Existence: with $A_n$ as above, $M_n = X_n - A_n$ satisfies $E[M_n - M_{n-1} \mid \mathcal{F}_{n-1}] = E[X_n \mid \mathcal{F}_{n-1}] - X_{n-1} - (A_n - A_{n-1}) = 0$. Uniqueness: if $M_n + A_n = M'_n + A'_n$, then $M - M'$ is a martingale that is predictable, so $M_n - M'_n = E[M_n - M'_n \mid \mathcal{F}_{n-1}] = M_{n-1} - M'_{n-1}$, and it is constant equal to $0$. Example: for the simple symmetric random walk $S_n$, $S_n^2 = (S_n^2 - n) + n$, so $A_n = n$. A sequence $(A_n)$ with $A_n$ $\mathcal{F}_{n-1}$-measurable is called predictable. Without the predictability of $A$ the decomposition is not unique: $X_n = 0 + X_n$ is another split into a martingale and an adapted process.

Going deeper

For a student

The predictable part $A$ of $M^2$ for a square-integrable martingale $M$ is the predictable quadratic variation $\langle M \rangle$, a basic tool for martingale central limit theorems and stochastic integration. The decomposition is the discrete-time model for the Doob–Meyer decomposition theorem, whose continuous-time proof is much harder. It is also the discrete form of the split of a semimartingale into a local martingale and a finite variation part.