the Doob-Meyer decomposition
/ Doob-Meyer = doob-MY-er /
In discrete time, any submartingale (X_n) splits uniquely as X_n = M_n + A_n with M a martingale and A a predictable increasing process starting at 0 — the Doob decomposition, where A_n - A_{n-1} = E[X_n given F_{n-1}] - X_{n-1} >= 0 is the 'expected upward push'. The Doob-Meyer decomposition is the (much harder) continuous-time analogue, and it is the structural theorem that makes the angle-bracket process, the compensator of a jump measure, and the whole Ito-integration program possible. It answers: can a continuous-time submartingale always be written as martingale plus increasing trend, with the trend canonically determined?
The theorem: let (X_t) be a cadlag submartingale satisfying a mild regularity condition (the original 'class D' condition, that {X_T : T a finite stopping time} is uniformly integrable; for the locally-bounded case one localizes). Then X has a unique decomposition X_t = M_t + A_t where M is a cadlag martingale, A is a predictable increasing process with A_0 = 0, and the decomposition is unique up to indistinguishability. The predictability of A is the crux — without it the decomposition fails to be unique (you could shuffle jumps between M and A), and it is the demand that A be predictable, not merely adapted, that pins it down. A is called the compensator of X: it is the predictable increasing process whose increments 'compensate' X to a martingale, the continuous-time analogue of the cumulative conditional means.
Doob-Meyer is the source of the central objects of stochastic calculus. Applied to the submartingale M^2 for a square-integrable martingale M, it produces the predictable quadratic variation <M> via M_t^2 - <M>_t being a martingale. Applied to a counting process, it produces the compensator (e.g. lambda t for a Poisson process), turning N_t - A_t into a martingale. The honest caveats: the original proofs were technically formidable (Meyer's class-D condition, later simplified by Rao); predictability of A is indispensable and is exactly where continuous and jump cases differ; and the decomposition is for submartingales (or, after a sign flip, supermartingales) — a general process need not decompose this way at all, which is precisely why semimartingales (sums of a local martingale and a finite-variation process) are singled out as the integrators.
For a Poisson process N_t with rate lambda, N is a submartingale (it only increases on average). Its Doob-Meyer decomposition is N_t = (N_t - lambda t) + lambda t: the martingale part is the compensated Poisson process, and the predictable increasing compensator is the smooth deterministic A_t = lambda t.
The compensator lambda t is the predictable increasing trend that makes N a martingale once subtracted.
Uniqueness hinges entirely on demanding the increasing part be predictable, not merely adapted; drop predictability and the decomposition is no longer unique because jumps can be reassigned between the martingale and the increasing process.