The Doob-Meyer Decomposition

A martingale is the mathematical formalization of a "fair game": given everything known so far, its expected future value equals its current value — no predictable drift in either direction. Many real processes aren't quite fair games; they drift upward or downward in a way that's foreseeable step by step, like a submartingale, whose expected next value is at least as large as the current one. The Doob-Meyer decomposition says that any such process, under mild technical conditions, can be split uniquely into two pieces: a martingale part, which is the genuinely unpredictable noise, and a "compensator," a predictable process that carries all of the systematic drift.

Intuitively, this separates "what happens because of new, unforeseeable information" from "what was already baked in and foreseeable given the past" — the compensator is built entirely from information already available, so it never surprises you, while the martingale part is the only piece that can move the process in a genuinely unexpected direction at each step. This underlies, for instance, the idea that a stock price with drift can be written as a driftless martingale plus a foreseeable drift term, and it is the theoretical backbone behind quadratic variation and stochastic integration more broadly, since many constructions in continuous-time finance rely on being able to isolate the "pure noise" component of a process cleanly.