Submartingales and Supermartingales from Conditional Drift

Conditional Drift and the Direction of a Process

A martingale has zero conditional expected increment: given the information available now, its expected next value equals its current value. Submartingales and supermartingales relax that equality in one direction. Their conditional expected increments are, respectively, nonnegative and nonpositive. The distinction is conditional: an increment that averages upward across all outcomes can still have a negative conditional expectation for some information already observed.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), in discrete time \(n\ge0\), with filtration \((\mathcal{F}_n)_{n\ge0}\). A process \((X_n)_{n\ge0}\) is adapted if \(X_n\) is \(\mathcal{F}_n\)-measurable for every \(n\). Assume throughout that each \(X_n\) is integrable, meaning \(\mathbb{E}[|X_n|]<\infty\). These assumptions make the conditional expectations below well-defined. Define the conditional drift at time \(n\) by

Because \(X_n\) is \(\mathcal{F}_n\)-measurable, the conditional-increment criterion for martingales gives the equality condition \(\mathbb{E}[X_{n+1}\mid\mathcal{F}_n]=X_n\) almost surely. The same decomposition identifies the one-sided versions.

Why the Drift Criterion Is Equivalent

The inequalities in the definition compare the next conditional value with the current value. Conditional drift instead describes the expected change. Under adaptation and integrability, these are two forms of the same condition.

Proof. Fix any \(n\). Linearity of conditional expectation and the \(\mathcal{F}_n\)-measurability of \(X_n\) give \(\Delta_n=\mathbb{E}[X_{n+1}\mid\mathcal{F}_n]-\mathbb{E}[X_n\mid\mathcal{F}_n]=\mathbb{E}[X_{n+1}\mid\mathcal{F}_n]-X_n\) almost surely. Thus \(\Delta_n\ge0\) almost surely if and only if \(\mathbb{E}[X_{n+1}\mid\mathcal{F}_n]\ge X_n\) almost surely. Replacing both inequalities by \(\le\) proves the supermartingale equivalence; replacing them by equality proves the martingale equivalence. Applying these equivalences at every \(n\) establishes the criteria for the whole process. \(\square\)

The sign condition is required at every time and almost surely. A conditional expectation is defined only up to changes on probability-zero events, so its inequality is interpreted almost surely, not as a pointwise claim for every \(\omega\). The drift may vary with both time and the information in \(\mathcal{F}_n\); it need not be a fixed number.

Proof. By the defining property of conditional expectation, \(\mathbb{E}[\Delta_n]=\mathbb{E}[X_{n+1}-X_n]\), which is the stated identity. If \(\Delta_n\ge0\) almost surely, its expectation is nonnegative, so the expected value cannot decrease; the nonpositive case gives the reverse conclusion. For strictness, suppose \(\Delta_n>0\) almost surely. The events \(\{\Delta_n\ge 1/k\}\), for positive integers \(k\), have union \(\{\Delta_n>0\}\). Since this union has probability one, at least one of these events has positive probability. For that \(k\), \(\mathbb{E}[\Delta_n]\ge \mathbb{P}(\Delta_n\ge1/k)/k>0\). The negative case follows by applying the same argument to \(-\Delta_n\). \(\square\)

Thus a nonnegative conditional drift gives a submartingale, but the expected value increases strictly only if the drift is positive on a set of positive probability. If the drift is zero almost surely, the expected value is unchanged at that step. Equality of unconditional expectations alone does not imply zero conditional drift: positive and negative conditional changes can cancel when averaged.

Constant and State-Dependent Drift

A particularly direct case is a constant conditional drift \(c\), meaning \(\Delta_n=c\) almost surely at each time under consideration. If \(c>0\), the process is a submartingale and its expectation rises by \(c\) per step. If \(c<0\), it is a supermartingale and its expectation falls by \(|c|\) per step. If \(c=0\), it is a martingale. The classification follows directly from the drift criterion; it does not require the realized increment \(X_{n+1}-X_n\) itself to equal \(c\).

Conditional drift can depend on the observed state. For a supermartingale, it is enough that every such conditional drift be nonpositive; the values need not be equal across states or times.

A Common Mistake: Using Unconditional Drift

The mean of an increment averages over all outcomes, including outcomes distinguished by information already available at time \(n\). The submartingale condition instead tests the expected increment given that information. These can have different signs on different branches.

Using the Classification Carefully

To classify a process from conditional drift, first specify the filtration: it determines what is known when the increment is evaluated. Then verify adaptation and integrability, compute \(\mathbb{E}[X_{n+1}-X_n\mid\mathcal{F}_n]\), and establish its sign almost surely for every relevant \(n\). A drift that is nonnegative at some times and nonpositive at others does not by itself give either classification; the required sign must hold throughout the time range.

The actual increments may fluctuate in either direction. What matters is their conditional mean given the present information. Conversely, a process with increasing unconditional expectations need not be a submartingale, as the two-branch example shows. These distinctions are useful whenever a process is built from evolving information: conditional drift gives a local criterion, while expected-value monotonicity is a consequence of that criterion rather than a replacement for it.