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.