The Doob–Meyer Decomposition: Predictability Depends on Information

Trend extraction feels intuitive. You look at a time series and ask: what part is structure, and what part is randomness?

To interpret structure as predictable drift, you first need to specify how information arrives over time.

For suitable submartingales, the Doob–Meyer theorem gives a unique split into a martingale and a predictable increasing process. This is a precise way to separate conditional-mean drift from a remainder; it is not a universal definition of signal and noise.

The split is relative to both a probability law and a filtration, the information available at each time. Its interpretation changes when those modeling choices change.

Specifying Predictable Drift

Let be an adapted stochastic process on a filtered probability space .

A naïve decomposition

does not specify which part represents predictable drift. Other definitions of signal, such as a latent physical quantity, answer different questions.

In discrete time, for an integrable adapted process and integer , one can write:

This identity separates a conditional mean from a prediction error:

The first term depends on the probability law and the information available at time .
The second term is only “noise” relative to that same information set.

This one-step prediction identity is different from a cumulative martingale decomposition. The prediction errors are martingale differences; their partial sums form a martingale.

Martingales and Zero Conditional-Mean Drift

A martingale formalizes the idea of zero conditional-mean drift.

A process is a martingale if:

Equivalently, the increment satisfies:

Assuming adaptation and integrability, this says that future increments have zero conditional mean. It does not make them independent of the past or prevent predictable changes in variance.

For example, let , where bounded is known at time and is independent of , has mean zero and variance one, and is observed at time . Then is a martingale, while the conditional variance of its next increment is . Volatility can be predictable even when mean drift is zero.

The Doob–Meyer Decomposition (Formal Statement)

On a finite horizon , let be a càdlàg submartingale of class D with respect to a complete, right-continuous filtration. Class D means that the family is uniformly integrable. Then there exists a unique decomposition:

such that:

is a uniformly integrable martingale.
is a predictable, càdlàg, increasing process with and .
The decomposition is unique up to indistinguishability.

The submartingale assumption makes the compensator increasing. Predictability is a measurability condition relative to the filtration; it does not mean the entire future path is known in advance. See Beiglböck, Schachermayer and Veliyev, A Short Proof of the Doob–Meyer Theorem.

Drift as Conditional Structure

The increasing process is the compensator that makes a martingale.

In discrete time, the related Doob decomposition applies to any integrable adapted process:

and

Each increment of is the one-step-ahead conditional expectation of the increment of . It is nonnegative when is a submartingale; for a general process, need not increase.

This makes the interpretation precise:

: accumulated one-step conditional-mean increments.
: the initial value plus accumulated conditional-mean-zero increments.

Nothing in this definition refers to smoothness, frequency, or parametric form. Only predictability matters.

Why Drift Is Not Intrinsic

In the discrete-time formula, is -measurable. Changing the information set can change the conditional expectations and therefore the decomposition.

Suppose and is equally likely to be or . With no initial information, its predictable increment is zero. If is revealed at time zero, that increment is itself and the martingale increment vanishes. Relative to the enlarged filtration, however, is no longer a submartingale on the event .

This example shows why a change of filtration requires checking the hypotheses again. Residual structure can also reflect an incorrect conditional-mean model using information already available, or predictable variance compatible with a martingale remainder. It does not by itself prove that the filtration is too small.

Relation to Filtering and State-Space Models

In a state-space model, distinguish the unobserved state from the observation filtration . Introducing a latent variable does not make it observable.

The observation innovation is:

It is a martingale difference with respect to the observation filtration. In a correctly specified linear Gaussian model, the Kalman filter computes the required conditional state means and covariances exactly. This connects filtering to conditional prediction; it is not, by itself, an algorithm for the increasing compensator in Doob–Meyer.

For the filtering equations, see Särkkä and Svensson, Bayesian Filtering and Smoothing.