# Doob decomposition theorem

The Doob decomposition theorem states that every integrable, adapted stochastic process in discrete time can be written, in an almost surely unique way, as the sum of a martingale and a [predictable process](https://www.edgepedia.org/predictable-process), and that the process is a submartingale exactly when the predictable part is increasing. Joseph L. Doob established the result in his 1953 book *Stochastic Processes*.<sup>[1](https://www.sciencedirect.com/topics/engineering/submartingales)</sup><sup> • </sup><sup>[2](https://obnb.uk/p8314296-stochastic-processes)</sup>

| Key fact | Detail |
|---|---|
| Statement | Any adapted, integrable process \( X \) satisfies \( X_n = X_0 + M_n + A_n \) with \( M \) a martingale and \( A \) predictable, \( M_0 = A_0 = 0 \); the pair \( (M, A) \) is almost surely unique.<sup>[3](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)</sup> |
| Submartingale test | \( X \) is a submartingale if and only if \( A \) is increasing, and a supermartingale if and only if \( A \) is decreasing.<sup>[3](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)</sup><sup> • </sup><sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/17%3A_Martingales/17.02%3A_Properties_and_Constructions)</sup> |
| Explicit compensator | \( A_n = \sum_{k=1}^{n} \bigl( \mathbb{E}[X_k \mid \mathcal{F}_{k-1}] - X_{k-1} \bigr) \), so each increment of \( A \) is the one-step conditional drift of \( X \).<sup>[5](https://www2.im.uj.edu.pl/MarcinPitera/files/StochasticProcesses.pdf)</sup> |
| Origin | Doob, *Stochastic Processes* (Wiley, 1953, 654 pages).<sup>[2](https://obnb.uk/p8314296-stochastic-processes)</sup> |
| Continuous-time analog | The Doob–Meyer decomposition: a càdlàg submartingale of class D decomposes uniquely as \( S = M + A \) with \( A \) predictable and increasing; due to Paul-André Meyer, with the modern formulation by Catherine Doléans-Dade.<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC4459556/)</sup> |
| Main application | Applied to \( M^2 \) for a square-integrable martingale \( M \), it produces the predictable quadratic variation \( \langle M \rangle \).<sup>[7](https://www.stats.ox.ac.uk/~etheridg/martingales.pdf)</sup> |

## Statement and construction

Fix a filtered probability space \( (\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_n)_{n \ge 0}) \), where the filtration \( (\mathcal{F}_n) \) records the information available at time \( n \). A process is adapted when \( X_n \) is \( \mathcal{F}_n \)-measurable, and predictable when \( A_n \) is already \( \mathcal{F}_{n-1} \)-measurable, that is, known one step in advance. A martingale is an adapted integrable process whose conditional expectation of the next value given the present information equals the current value.

**The theorem.** Let \( X \) be an adapted, integrable process. There exist a martingale \( M \) and a predictable process \( A \) with \( A_0 = M_0 = 0 \) such that

\[ X_n = X_0 + M_n + A_n, \qquad n \ge 0, \]

and the pair \( (M, A) \) is almost surely unique. The process \( X \) is a submartingale if and only if \( A \) is increasing, and a supermartingale if and only if \( A \) is decreasing.<sup>[3](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)</sup><sup> • </sup><sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/17%3A_Martingales/17.02%3A_Properties_and_Constructions)</sup>

The construction is explicit. Define the predictable drift increments and accumulate them:

\[ A_n = \sum_{k=1}^{n} \bigl( \mathbb{E}[X_k \mid \mathcal{F}_{k-1}] - X_{k-1} \bigr), \qquad M_n = X_n - X_0 - A_n. \]

Each summand is \( \mathcal{F}_{k-1} \)-measurable by the definition of conditional expectation, so \( A \) is predictable. Taking conditional expectations of the increments of \( M \) shows that \( \mathbb{E}[M_k - M_{k-1} \mid \mathcal{F}_{k-1}] = 0 \), so \( M \) is a martingale.<sup>[5](https://www2.im.uj.edu.pl/MarcinPitera/files/StochasticProcesses.pdf)</sup> In the common convention for a submartingale \( x \) indexed from 0, the decomposition is written \( x_n = M_n + A_n \) with \( A_{n+1} \ge A_n \) and \( A_n \in \mathcal{F}_{n-1} \) (setting \( \mathcal{F}_{-1} := \mathcal{F}_0 \)).<sup>[1](https://www.sciencedirect.com/topics/engineering/submartingales)</sup>

**Uniqueness.** If \( X = M + A = M' + A' \) are two decompositions, then \( M - M' = A' - A \) is simultaneously a martingale and a predictable process. A predictable martingale is constant, because its next increment has conditional expectation zero yet is already known; starting from \( A_0 = A'_0 = 0 \), induction gives \( A_n = A'_n \) and \( M_n = M'_n \) for all \( n \), almost surely.<sup>[3](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)</sup><sup> • </sup><sup>[8](https://adityam.github.io/stochastic-control/probability/martingales.html)</sup>

The monotonicity clause is what gives the theorem its diagnostic value. Since \( A_n - A_{n-1} = \mathbb{E}[X_n - X_{n-1} \mid \mathcal{F}_{n-1}] \), the process \( X \) is a submartingale precisely when these conditional drifts are nonnegative, which is exactly the statement that \( A \) increases.<sup>[3](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)</sup> Without any monotonicity requirement on \( A \), the decomposition exists for every integrable adapted discrete-time process and remains almost surely unique.<sup>[5](https://www2.im.uj.edu.pl/MarcinPitera/files/StochasticProcesses.pdf)</sup>

## Interpretation: drift plus martingale noise

The decomposition separates a process into the part that is conditionally predictable one step ahead and the part that is conditionally unpredictable. The martingale part \( M \) has increments \( X_k - \mathbb{E}[X_k \mid \mathcal{F}_{k-1}] \), the surprises of \( X \); the predictable part \( A \) accumulates the conditional drifts. In financial terms, \( A \) is the predictable drift and \( M \) the martingale noise of a discounted asset under the physical measure.<sup>[9](https://wanghemath.github.io/Book-MathFinance/chapters/chapter-05-discrete-martingales.html)</sup>

For a random walk \( M_n = \sum_{i=1}^{n} \eta_i \) of independent centered increments, the martingale part is the walk itself and \( A \equiv 0 \); if the increments have nonzero mean, that mean accumulates in \( A \).<sup>[8](https://adityam.github.io/stochastic-control/probability/martingales.html)</sup> For an Euler discretization of a stochastic differential equation \( dX_t = b(X_t)\,dt + \sigma(X_t)\,dB_t \), the Doob decomposition has \( A^{(h)}_n = \sum_{k=0}^{n-1} b(X^{(h)}_k) \cdot h \) and \( M^{(h)}_n = x_0 + \sum_{k=0}^{n-1} \sigma(X^{(h)}_k) \sqrt{h}\, \eta_{k+1} \), so the predictable part discretizes the drift vector field and the martingale part the noise.<sup>[10](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1516/IntroStochAnalysis2015.pdf)</sup>

## Consequences and applications

**Quadratic variation.** If \( M \) is a square-integrable martingale, [Jensen's inequality](https://www.edgepedia.org/jensens-inequality) makes \( M^2 \) a submartingale, so the Doob decomposition of \( M^2 \) reads \( M_n^2 = \tilde{M}_n + \langle M \rangle_n \), where \( \langle M \rangle \) is a predictable increasing process called the (predictable) quadratic variation or conditional variance process. Its increments are

\[ \langle M \rangle_k - \langle M \rangle_{k-1} = \mathbb{E}\bigl[ (M_k - M_{k-1})^2 \bigm| \mathcal{F}_{k-1} \bigr] = \operatorname{Var}\bigl[ M_k - M_{k-1} \bigm| \mathcal{F}_{k-1} \bigr], \]

and \( \mathbb{E}[M_n^2] = \mathbb{E}[M_0^2] + \mathbb{E}[\langle M \rangle_n] \), a basic tool for \( L^2 \) estimates. For a random walk of independent increments, \( \langle M \rangle_n = \sum_{i=1}^{n} \operatorname{Var}[\eta_i] \).<sup>[7](https://www.stats.ox.ac.uk/~etheridg/martingales.pdf)</sup><sup> • </sup><sup>[10](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1516/IntroStochAnalysis2015.pdf)</sup> The ordinary quadratic variation \( [M]_n = \sum_{k=1}^{n} (M_k - M_{k-1})^2 \) also satisfies that \( M_n^2 - [M]_n \) is a martingale, but \( [M] \) is not predictable; in discrete time the two processes usually differ.<sup>[10](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1516/IntroStochAnalysis2015.pdf)</sup>

**Maximal inequalities.** Combining the decomposition with the submartingale inequality yields bounds on running maxima: for a submartingale or supermartingale \( X \), \( \lambda\, \mathbb{P}(\max_{k \le n} |X_k| \ge 3\lambda) \le 4\,\mathbb{E}|X_0| + 3\,\mathbb{E}|X_n| \). The proof splits \( X \) into \( X_0 \), the martingale \( M \), and the increasing \( A \), and bounds each term separately.<sup>[3](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)</sup>

**Markov chains and the martingale problem.** For a time-homogeneous [Markov chain](https://www.edgepedia.org/markov-chain) with transition kernel \( p \), the Doob decomposition of \( f(X_n) \) has predictable part \( \sum_{k=0}^{n-1} (\mathcal{L}f)(X_k) \) with generator \( \mathcal{L} = p - I \). Requiring this decomposition to hold with a martingale part for every bounded \( f \) is equivalent to \( (X_n) \) being a Markov chain with kernel \( p \), which is the discrete-time martingale problem. Harmonic functions (\( \mathcal{L}f = 0 \)) make \( f(X_n) \) a martingale, and superharmonic functions make it a supermartingale.<sup>[10](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1516/IntroStochAnalysis2015.pdf)</sup>

## The continuous-time analog: Doob–Meyer

Doob's discrete-time result extends to continuous time under additional regularity. The [Doob–Meyer decomposition theorem](https://www.edgepedia.org/doob-meyer-decomposition-theorem) states that a càdlàg submartingale \( S \) of class D, meaning that the family \( \{S_\tau\} \) over all stopping times \( \tau \) is uniformly integrable, can be written uniquely as \( S = M + A \) with \( M \) a martingale and \( A \) a predictable increasing process starting at 0.<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC4459556/)</sup>

The history runs through several steps. Meyer introduced class D and proved the decomposition with \( A \) a natural process; Doléans-Dade then showed that an increasing process is natural if and only if it is predictable, giving the modern formulation. Further proofs were given by Rao, who obtained the continuous-time decomposition as a weak-\( L^1 \) limit of discrete approximations, by Bass, and by Jakubowski.<sup>[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC4459556/)</sup> The discrete theorem itself needs none of this analytic machinery: existence and uniqueness follow from the explicit conditional-expectation formula alone.<sup>[5](https://www2.im.uj.edu.pl/MarcinPitera/files/StochasticProcesses.pdf)</sup>

In continuous time the decomposition underlies the compensator of a counting process in survival analysis and the predictable quadratic variation \( \langle M \rangle \) of square-integrable martingales, the sharp bracket of stochastic calculus.<sup>[11](https://web.stanford.edu/~lutian/coursepdf/survweek6.pdf)</sup><sup> • </sup><sup>[10](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1516/IntroStochAnalysis2015.pdf)</sup>

## Related decompositions

Variants of the theorem include the Riesz and Krickenberg decompositions of submartingales.<sup>[12](https://www.stat.cmu.edu/~aramdas/martingales18/L2-martingales.pdf)</sup> The decomposition also extends beyond the classical setting: optional Doob decompositions have been proved for local regular supermartingales relative to a convex set of equivalent measures, and Doob–Meyer-type decompositions have been developed for \( g \)-supermartingale systems and nonlinear expectations arising from quadratic [backward stochastic differential equations](https://www.edgepedia.org/backward-stochastic-differential-equation).<sup>[13](https://arxiv.org/abs/1601.03574)</sup><sup> • </sup><sup>[14](https://arxiv.org/abs/1505.00597)</sup>

## References

1. [Doob's decomposition, ScienceDirect Topics: Submartingales (Poznyak, *Advanced Mathematical Tools for Automatic Control Engineers*, Vol. 2, 2009)](https://www.sciencedirect.com/topics/engineering/submartingales)
2. [Stochastic processes, Joseph Leo Doob, Wiley 1953, OBNB catalogue record](https://obnb.uk/p8314296-stochastic-processes)
3. [Lecture notes on martingales, Theorem 2.2.8 (Doob decomposition), Duke University](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)
4. [The Doob Decomposition, Statistics LibreTexts (Siegrist), Properties and Constructions](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/17%3A_Martingales/17.02%3A_Properties_and_Constructions)
5. [Stochastic Processes lecture notes, Theorem 2.41 (Doob decomposition), Jagiellonian University](https://www2.im.uj.edu.pl/MarcinPitera/files/StochasticProcesses.pdf)
6. [Beiglböck, Schachermayer, Veliyev, "A short proof of the Doob–Meyer theorem", PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC4459556/)
7. [Martingales lecture notes, University of Oxford (Etheridge)](https://www.stats.ox.ac.uk/~etheridg/martingales.pdf)
8. [Martingales, Stochastic Control and Decision Theory, Theorem 49.1](https://adityam.github.io/stochastic-control/probability/martingales.html)
9. [Chapter 5, Martingales in Discrete Time, Mathematical Finance](https://wanghemath.github.io/Book-MathFinance/chapters/chapter-05-discrete-martingales.html)
10. [Eberle, Introduction to Stochastic Analysis, University of Bonn, Chapter 2](https://wt.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Andreas_Eberle/IntroStoAn1516/IntroStochAnalysis2015.pdf)
11. [Survival Analysis: Counting Process and Martingale, Stanford](https://web.stanford.edu/~lutian/coursepdf/survweek6.pdf)
12. [Martingales lecture notes, Carnegie Mellon University (A. Ramdas)](https://www.stat.cmu.edu/~aramdas/martingales18/L2-martingales.pdf)
13. [Generalization of Doob decomposition Theorem, arXiv:1601.03574](https://arxiv.org/abs/1601.03574)
14. [A general Doob-Meyer-Mertens decomposition for g-supermartingale systems, arXiv:1505.00597](https://arxiv.org/abs/1505.00597)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Martingales and filtrations › Martingale transforms and discrete martingale calculus*

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