Martingale Theory

Martingales, submartingales and supermartingales: definitions, examples and basic properties.
Author

John Robin Inston

Published

September 25, 2026

1 Martingales

A martingale models a fair game: given everything known so far, the expected future value equals the present value. Let \((\Omega,\mathcal{F},\mathbb{F},\mathbb{P})\) be a probability space with a filtration \(\mathbb{F}=(\mathcal{F}_t)_{t\in T}\), where the time set \(T\) is \(\{0,1,2,\dots\}\) in discrete time or \([0,\infty)\) in continuous time.

A stochastic process \(X=(X_t)_{t\in T}\) is a martingale with respect to \(\mathbb{F}\) if

  1. \(X\) is adapted to \(\mathbb{F}\), that is \(X_t\) is \(\mathcal{F}_t\)-measurable for all \(t\);
  2. \(\mathbb{E}[|X_t|]<\infty\) for all \(t\); and
  3. \(\mathbb{E}[X_t \mid \mathcal{F}_s] = X_s\) for all \(s \le t\).

In discrete time it is enough to check the one-step condition \(\mathbb{E}[X_{n+1}\mid\mathcal{F}_n] = X_n\) for all \(n\), since the general case follows from the tower property. In continuous time one also assumes that the paths are right-continuous with left limits and that the filtration satisfies the usual conditions.

An adapted, integrable process \(X\) is a

  • submartingale if \(\mathbb{E}[X_t\mid\mathcal{F}_s] \ge X_s\) for all \(s\le t\) (the process tends to drift upwards in expectation),
  • supermartingale if \(\mathbb{E}[X_t\mid\mathcal{F}_s] \le X_s\) for all \(s\le t\) (the process tends to drift downwards in expectation).

A process is a martingale if and only if it is both a submartingale and a supermartingale. Think of \(X_t\) as the wealth of a gambler: a martingale is a fair game, a submartingale is a favorable game and a supermartingale is an unfavorable game.

2 Examples

  1. Random walk. If \(X_1,X_2,\dots\) are independent with \(\mathbb{E}[X_i]=0\), then \(S_n=X_1+\dots+X_n\) is a martingale. If \(\mathbb{E}[X_i]\ge 0\) it is a submartingale.
  2. Doob martingale. For an integrable random variable \(Y\), \(M_n=\mathbb{E}[Y\mid\mathcal{F}_n]\) is a martingale, since by the tower property \(\mathbb{E}[M_{n+1}\mid\mathcal{F}_n]=\mathbb{E}[Y\mid\mathcal{F}_n]=M_n\).
  3. Products. If \(X_1,X_2,\dots\) are independent with \(\mathbb{E}[X_i]=1\), then \(\prod_{i\le n}X_i\) is a martingale. For example, with \(S_n\) as in (1) and \(\mathbb{E}[e^{\theta X_1}]<\infty\), \(e^{\theta S_n}/\mathbb{E}[e^{\theta X_1}]^n\) is a martingale.
  4. Brownian motion. If \(W\) is a Brownian motion, then \(W_t\), \(W_t^2-t\) and \(\exp(\theta W_t-\tfrac12\theta^2t)\) are martingales.
  5. Discounted prices. In the Black-Scholes-Merton Model, the discounted stock price \(e^{-rt}S_t\) is a martingale under the risk-neutral measure \(\mathbb{Q}\). See Risk-Neutral Pricing.

3 Basic Properties

  • Constant expectation. For a martingale, \(\mathbb{E}[X_t]=\mathbb{E}[X_0]\) for all \(t\). For a submartingale, \(t\mapsto\mathbb{E}[X_t]\) is nondecreasing.
  • Jensen. If \(X\) is a martingale and \(\varphi\) is convex with \(\varphi(X_t)\) integrable, then \(\varphi(X)\) is a submartingale, since by the conditional Jensen inequality \(\mathbb{E}[\varphi(X_{t})\mid\mathcal{F}_s] \ge \varphi(\mathbb{E}[X_t\mid\mathcal{F}_s]) = \varphi(X_s)\). In particular \(|X|\) and \(X^2\) (when \(X\) is square integrable) are submartingales.
  • Martingale differences. The increments of a martingale satisfy \(\mathbb{E}[X_{n+1}-X_n\mid\mathcal{F}_n]=0\). See Martingale Differences.
  • Martingale transforms. A process \(H\) is predictable if \(H_n\) is \(\mathcal{F}_{n-1}\)-measurable. For a bounded predictable \(H\) the transform \[ (H\bullet X)_n = \sum_{k=1}^n H_k\,(X_k-X_{k-1}) \] is a martingale whenever \(X\) is a martingale, and a submartingale if \(X\) is a submartingale and \(H\ge 0\). This is because \(\mathbb{E}[H_k(X_k-X_{k-1})\mid\mathcal{F}_{k-1}]=H_k\,\mathbb{E}[X_k-X_{k-1}\mid\mathcal{F}_{k-1}]\). It says that no betting strategy turns a fair game into a favorable one.
  • Stopped processes. If \(\tau\) is a stopping time and \(X\) is a martingale, then the stopped process \(X_{\tau\wedge n}\) is a martingale. This is the transform with \(H_k=\mathbb{1}_{\{k\le\tau\}}\).

4 Key Results

The main theorems of martingale theory each have their own note.

Planned notes: local martingales and semimartingales, quadratic variation of martingales, exponential martingales and Novikov’s condition, Lévy’s characterization of Brownian motion, Azuma-Hoeffding concentration, and applications (gambler’s ruin, Pólya’s urn, branching processes).

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