---
tag: SX7ZPP72EM
kind: statement
role: theorem
name: Doob–Meyer decomposition theorem
msc: [60G44, 60G48]
uses: [EQ055QMACW, 0CDVK9RQBH, HNK1Y62G7V, WND87FBAKC, YRNFE016FF]
about: [6M852B4RA8, 0D0GBNDKDH]
sources:
  - url: https://en.wikipedia.org/wiki/Doob%E2%80%93Meyer_decomposition_theorem
same_as:
  wikidata: Q5296972
  wikipedia: Doob–Meyer decomposition theorem
---

## Statement

Let $(\Omega, \mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\mathcal{F}_t)_{t \ge 0}$ that is right-continuous and complete.
Let $(X_t)_{t \ge 0}$ be a [[submartingale|EQ055QMACW]] with càdlàg paths, that is, right-continuous paths with left limits.
Assume $X$ is of class (D): the family of random variables $X_\tau$, where $\tau$ runs over the finite [[stopping times|WND87FBAKC]], is uniformly integrable.
Then there is a unique decomposition $X_t = M_t + A_t$ for $t \ge 0$ with these properties:

1. $(M_t)$ is a uniformly integrable càdlàg [[martingale|0CDVK9RQBH]].
2. $(A_t)$ is a predictable, non-decreasing càdlàg process with $A_0 = 0$.
3. $E[A_\infty] < \infty$, where $A_\infty = \lim_{t \to \infty} A_t$.

Uniqueness is up to indistinguishability.

## Remarks

Here a process is predictable if it is measurable for the σ-algebra on $[0, \infty) \times \Omega$ generated by the left-continuous [[adapted|YRNFE016FF]] processes.
One proof applies the discrete [[Doob decomposition theorem|YHSSQ67GDS]] on dyadic time grids and shows that the predictable parts converge, using class (D) for uniform integrability.
Example: for a Wiener process $W$ and $T > 0$, the stopped process $X_t = W_{t \wedge T}^2$ is a submartingale of class (D), and its decomposition is $M_t = W_{t \wedge T}^2 - t \wedge T$, $A_t = t \wedge T$.
Without class (D) a decomposition still exists locally, with $M$ a [[local martingale|ZWJ9G89W3T]]; this is used for $W_t^2 = (W_t^2 - t) + t$ on $[0, \infty)$.
Predictability is what makes $A$ unique.
For a Poisson process $N$ with rate $1$, both $N_t = 0 + N_t$ and $N_t = (N_t - t) + t$ split $N$ into a martingale and an increasing process.
Only $A_t = t$ is predictable.

## Explanations

### L1 human

A process that tends to rise splits into a fair game plus a rising trend.
The trend can be foreseen an instant before it happens, and the fair game holds all the surprise.
This is the continuous-time version of the Doob decomposition for sequences.

### L2 human

The theorem is the source of the predictable quadratic variation $\langle M \rangle$ of a square-integrable martingale $M$, the compensator $A$ of $M^2$.
It underlies the construction of the stochastic integral with respect to martingales and the theory of [[semimartingales|MZNFPJJMY9]].
Compensators of counting processes, such as $t$ for the Poisson process, come from the same theorem and are central in point process theory.