Martingales and Submartingales¶
A martingale is the mathematical formalization of a fair game: given all past information, the expected future value equals the current value. The name originated with an 18th-century doubling-up betting strategy; the modern concept, due to Doob, is far more general and underlies the pricing of derivatives, the analysis of stochastic algorithms, and the characterization of stochastic noise.
Definition¶
Let \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})\) be a filtered probability space. An adapted process \(M = (M_t)_{t \ge 0}\) is a martingale if:
- Integrability: \(\mathbb{E}|M_t| < \infty\) for all \(t \ge 0\).
- Martingale property: for all \(0 \le s \le t\),
Equivalently, \(\mathbb{E}[M_t - M_s \mid \mathcal{F}_s] = 0\): increments are uncorrelated with the past (though not necessarily independent).
Interpretation
Given the information \(\mathcal{F}_s\) up to time \(s\), the best prediction of \(M_t\) is \(M_s\). The process has no predictable drift.
Sub- and Supermartingales¶
Relaxing the equality in the definition yields two generalizations:
- Submartingale: \(\mathbb{E}[X_t \mid \mathcal{F}_s] \ge X_s\) — favorable game, upward drift.
- Supermartingale: \(\mathbb{E}[X_t \mid \mathcal{F}_s] \le X_s\) — unfavorable game, downward drift.
The naming follows potential theory: supermartingales dominate their futures on average (like superharmonic functions), while submartingales are dominated by them.
Fundamental Examples¶
Brownian motion
Standard Brownian motion \(W_t\) is a martingale under its natural filtration: using properties of conditional expectation (see Conditional Expectation), for \(s \le t\),
since \(W_t - W_s\) is independent of \(\mathcal{F}_s\) with mean zero. \(\square\)
Compensated squared Brownian motion
\(M_t = W_t^2 - t\) is a martingale. Writing \(W_t = W_s + \Delta\) with \(\Delta = W_t - W_s \perp \mathcal{F}_s\), the cross term vanishes and \(\mathbb{E}[\Delta^2] = t-s\), giving \(\mathbb{E}[W_t^2 \mid \mathcal{F}_s] = W_s^2 + (t-s)\). Subtracting \(t\) makes this \(W_s^2 - s\). \(\square\)
Insight: \(W_t^2\) grows on average at rate 1; subtracting \(t\) removes this drift.
Doob martingale
For any \(X \in L^1(\mathcal{F})\), \(M_t := \mathbb{E}[X \mid \mathcal{F}_t]\) is a martingale. By the tower property, for \(s \le t\),
This is the canonical construction: \(X\) is revealed gradually as \(t\) increases.
Simple random walk
For i.i.d. \(\xi_k\) with \(\mathbb{P}(\xi_k = \pm 1) = 1/2\), the partial sum \(S_n = \sum_{k=1}^n \xi_k\) is a martingale under \(\mathcal{F}_n = \sigma(\xi_1, \ldots, \xi_n)\).
Convex Functions Produce Submartingales¶
Proposition
If \(M\) is a martingale and \(\varphi : \mathbb{R} \to \mathbb{R}\) is convex with \(\mathbb{E}|\varphi(M_t)| < \infty\), then \(\varphi(M)\) is a submartingale.
Proof. By conditional Jensen's inequality,
Consequence: \(|M_t|\), \(M_t^2\), \((M_t)^+\), and \((M_t)^-\) are all submartingales when \(M\) is a martingale. This fact drives Doob's maximal inequalities in the next section.
\(L^p\) Martingales and Uniform Integrability¶
A martingale \(M\) is \(L^p\)-bounded if \(\sup_t \mathbb{E}|M_t|^p < \infty\). Recall (see § Uniform Integrability): a family is UI if \(\lim_{K \to \infty} \sup_\alpha \mathbb{E}[|X_\alpha| \mathbf{1}_{\{|X_\alpha| > K\}}] = 0\); \(L^p\)-boundedness with \(p>1\) implies UI, \(L^1\)-boundedness alone does not, and UI is the hypothesis for \(L^1\) convergence and unbounded optional sampling.
Summary¶
| Concept | Definition |
|---|---|
| Martingale | \(\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s\) |
| Submartingale | \(\mathbb{E}[X_t \mid \mathcal{F}_s] \ge X_s\) |
| Supermartingale | \(\mathbb{E}[X_t \mid \mathcal{F}_s] \le X_s\) |
| Doob martingale | \(M_t = \mathbb{E}[X \mid \mathcal{F}_t]\) |
| Convex transform | \(\varphi(M)\) submartingale for \(M\) martingale, \(\varphi\) convex |
The martingale property — absence of predictable drift — is one of the most powerful structural assumptions in stochastic analysis. Subsequent sections develop the consequences: stopping times, optional sampling, Doob's maximal inequalities, martingale convergence, uniform integrability, and the Doob–Meyer decomposition.
Exercises¶
Exercise 1. For standard Brownian motion \(W\), classify each process as martingale, sub-, super-, or none.
(a) \(W_t^2 - t\)
(b) \(e^{W_t}\)
(c) \(W_t^3 - 3tW_t\)
(d) \(W_t^4 - 6tW_t^2 + 3t^2\)
Solution to Exercise 1
Let \(\Delta = W_t - W_s \sim N(0, t-s)\), independent of \(\mathcal{F}_s\), with moments \(0, t-s, 0, 3(t-s)^2\).
(a) Martingale. Shown in the main text: \(\mathbb{E}[W_t^2 - t \mid \mathcal{F}_s] = W_s^2 - s\).
(b) Strict submartingale. \(\mathbb{E}[e^{W_t} \mid \mathcal{F}_s] = e^{W_s}\mathbb{E}[e^{\Delta}] = e^{W_s + (t-s)/2} > e^{W_s}\).
(c) Martingale. Expand \(W_t^3 = W_s^3 + 3W_s^2\Delta + 3W_s\Delta^2 + \Delta^3\) and take conditional expectation: \(\mathbb{E}[W_t^3 \mid \mathcal{F}_s] = W_s^3 + 3(t-s)W_s\). Hence \(\mathbb{E}[W_t^3 - 3tW_t \mid \mathcal{F}_s] = W_s^3 + 3(t-s)W_s - 3tW_s = W_s^3 - 3sW_s\).
(d) Martingale (fourth Hermite polynomial). Expand \((W_s + \Delta)^4\) and use the moments of \(\Delta\): \(\mathbb{E}[W_t^4 \mid \mathcal{F}_s] = W_s^4 + 6W_s^2(t-s) + 3(t-s)^2\). Then \(\mathbb{E}[W_t^4 - 6tW_t^2 + 3t^2 \mid \mathcal{F}_s] = W_s^4 + 6W_s^2(t-s) + 3(t-s)^2 - 6t(W_s^2 + (t-s)) + 3t^2 = W_s^4 - 6sW_s^2 + 3s^2\).
Exercise 2. Let \(X \in L^1(\mathcal{F})\) and \(M_t = \mathbb{E}[X \mid \mathcal{F}_t]\).
(a) Prove \(M\) is a martingale.
(b) Show that if \(X = W_T\), then \(M_t = W_{t \wedge T}\).
(c) For \(X = W_T^2\) and \(t \le T\), find \(M_t\).
Solution to Exercise 2
(a) Tower property: \(\mathbb{E}[M_t \mid \mathcal{F}_s] = \mathbb{E}[\mathbb{E}[X \mid \mathcal{F}_t] \mid \mathcal{F}_s] = \mathbb{E}[X \mid \mathcal{F}_s] = M_s\) for \(s \le t\). \(\square\)
(b) For \(t \le T\): \(M_t = \mathbb{E}[W_T \mid \mathcal{F}_t] = W_t\) (Brownian martingale property). For \(t \ge T\): \(W_T\) is \(\mathcal{F}_T \subseteq \mathcal{F}_t\)-measurable, so \(M_t = W_T\). Hence \(M_t = W_{t \wedge T}\).
(c) For \(t \le T\): \(\mathbb{E}[W_T^2 \mid \mathcal{F}_t] = W_t^2 + (T - t)\).
Exercise 3. Let \(\xi_k\) be i.i.d. with \(\mathbb{P}(\xi_k = 1) = p\), \(\mathbb{P}(\xi_k = -1) = 1 - p\), and \(S_n = \sum_{k=1}^n \xi_k\).
(a) For what \(p\) is \(S_n\) a martingale?
(b) For general \(p\), find \(f\) such that \(S_n - n f(p)\) is a martingale.
(c) Show \(M_n = \left(\frac{1-p}{p}\right)^{S_n}\) is a martingale for any \(p \in (0,1)\).
Solution to Exercise 3
(a) \(\mathbb{E}[\xi_k] = 2p - 1 = 0 \Leftrightarrow p = 1/2\).
(b) With \(\mu = 2p - 1\), \(\mathbb{E}[S_{n+1} - S_n \mid \mathcal{F}_n] = \mu\), so \(S_n - n\mu\) has zero drift. Hence \(f(p) = 2p - 1\).
(c) Let \(r = (1-p)/p\). Independence of \(\xi_{n+1}\) from \(\mathcal{F}_n\) and "taking out what is known" give
So \(M_n\) is a martingale. \(\square\)
Exercise 4. Prove: if \(M\) is a martingale and \(\varphi\) is convex with \(\mathbb{E}|\varphi(M_t)| < \infty\), then \(\varphi(M)\) is a submartingale. Deduce that \(|W_t|\), \(W_t^2\), and \((W_t)^+\) are submartingales.
Solution to Exercise 4
By conditional Jensen's inequality: \(\mathbb{E}[\varphi(M_t) \mid \mathcal{F}_s] \ge \varphi(\mathbb{E}[M_t \mid \mathcal{F}_s]) = \varphi(M_s)\). Adaptedness and integrability hold by hypothesis. \(\square\)
Apply with \(\varphi(x) = |x|\), \(x^2\), \(x^+ = \max(x,0)\) respectively (all convex).
Exercise 5. Let \(M\) be a martingale on \([0, T]\) with \(\mathbb{E}[M_T^2] < \infty\). Show that \(\mathbb{E}[(M_t - M_s)^2] = \mathbb{E}[M_t^2] - \mathbb{E}[M_s^2]\) for \(s \le t\), and that \(t \mapsto \mathbb{E}[M_t^2]\) is nondecreasing.
Solution to Exercise 5
Expand \((M_t - M_s)^2 = M_t^2 - 2M_s M_t + M_s^2\). Take expectations, and condition the middle term on \(\mathcal{F}_s\): \(\mathbb{E}[M_s M_t] = \mathbb{E}[M_s \mathbb{E}[M_t \mid \mathcal{F}_s]] = \mathbb{E}[M_s^2]\) by the martingale property. Hence
The LHS is \(\ge 0\), so \(\mathbb{E}[M_s^2] \le \mathbb{E}[M_t^2]\). \(\square\)
Remark: this says martingale increments are orthogonal in \(L^2\), and \(t \mapsto \mathbb{E}[M_t^2]\) plays the role of "accumulated variance" — foreshadowing the quadratic variation \(\langle M \rangle_t\) of the Doob–Meyer decomposition.