Doob–Meyer decomposition theorem

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title: "Doob–Meyer decomposition theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["martingale-theory", "theorems-in-statistics", "theorems-in-probability-theory"] description: "Theorem in stochastic calculus" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Doob–Meyer_decomposition_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Theorem in stochastic calculus ::

The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. It is named for Joseph L. Doob and Paul-André Meyer.

History

In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales. He conjectured a continuous time version of the theorem and in two publications in 1962 and 1963 Paul-André Meyer proved such a theorem, which became known as the Doob-Meyer decomposition. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.

Class D supermartingales

A càdlàg supermartingale Z is of Class D if Z_0=0 and the collection : {Z_T \mid T \text{ a finite-valued stopping time} } is uniformly integrable.

Theorem

Let (\Omega, \mathcal{F},(\mathcal{F}t){t \ge 0}, \mathbb{P}) be a filtered probability space satisfying the usual conditions (i.e. the filtration is right-continuous and complete; see Filtration (probability theory)). If X = (X_t)_{t\ge0} is a right-continuous submartingale of class D, then there exist unique adapted processes M and A such that : X_t = M_t + A_t, \qquad t \ge 0, where

• M is a uniformly integrable martingale,
• A is a predictable, right-continuous, increasing process with A_0 = 0. The decomposition (M, A) is unique up to indistinguishability.

Remark. For a class D supermartingale, the process A is integrable and of finite variation on bounded intervals.

Notes

References

References

1. Doob 1953
2. Meyer 1962
3. Meyer 1963
4. Protter 2005
5. Protter (2005)
6. Karatzas & Shreve (1991), Theorem 4.10.

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