Martingales

A martingale is a process whose conditional expectation, given everything known so far, is its current value: a mathematical model of a fair game, and — more usefully — the general structure of "conditional expectation indexed by time".

Martingales are included here for three foundational reasons. They give a convergence theorem requiring neither independence nor identical distribution; they supply the machinery behind Lévy's zero–one law; and they are the setting in which sequential conditioning is unambiguous, the conditioning family being fixed in advance rather than chosen after the fact (conditional-expectation.md §6).

References: Williams, Probability with Martingales (the canonical treatment); Durrett, Probability, ch. 4; Kallenberg, Foundations, ch. 7.

1. Filtrations

Definition. A filtration is an increasing family ${\mathcal{F}}_0\subseteq >{\mathcal{F}}_1\subseteq \cdots \subseteq \mathcal{F}$ of sub-$\sigma$-algebras. A process $(X_n)$ is adapted if $X_n$ is ${\mathcal{F}}_n$-measurable for each $n$, and predictable if $X_n$ is ${\mathcal{F}}_{n-1}$-measurable.

${\mathcal{F}}_n$ is "what is known at time $n$" — the informational reading of a $\sigma$-algebra introduced in kolmogorov-axioms.md §2(b). The natural filtration of a process is ${\mathcal{F}}_n=\sigma (X_0,\dots ,X_n)$. Predictability is the formal version of "decided before seeing the next outcome", and is exactly what a gambling strategy must satisfy.

Definition. An adapted, integrable process $(X_n)$ is a

• martingale if $E[X_{n+1}\mid {\mathcal{F}}_n]=X_n$ a.s.;
• supermartingale if $E[X_{n+1}\mid {\mathcal{F}}_n]\le X_n$ (decreasing on average — the unfavourable game);
• submartingale if $\ge$.

By the tower property (conditional-expectation.md §3), $E[X_n\mid {\mathcal{F}}_m]=X_m$ for all $n\ge m$, and $E[X_n]=E[X_0]$ for every $n$: a martingale has constant mean.

Standard examples: sums $S_n={\sum }_{i\le n}{X}_i$ of independent centred variables; products of independent unit-mean positive variables; the Doob martingale $M_n=E[Y\mid {\mathcal{F}}_n]$ for any fixed $Y\in L^1$ (which by the tower property is automatically a martingale — the source of Lévy's law); and $S_n^2-n{\sigma }^2$ for a centred random walk.

2. Stopping Times and Optional Stopping

Definition. A random time $\tau :\Omega \to \{0,1,\dots \}\cup \{\infty \}$ is a stopping time if $\{\tau \le n\}\in {\mathcal{F}}_n$ for every $n$ — whether to stop at $n$ is decidable from information available at $n$.

"The first time the walk hits $10$" is a stopping time; "the last time it visits $0$" is not (it requires knowing the future). The distinction is the entire content of the next theorem.

Theorem (optional stopping). Let $(X_n)$ be a martingale and $\tau$ a stopping time. Then $E[X_{\tau }]=E[X_0]$ provided any one of:

1. $\tau$ is bounded;
2. $\tau <\infty$ a.s. and $(X_{n\wedge \tau })$ is uniformly integrable;
3. $E[\tau ]<\infty$ and the increments are bounded.

Proof idea. $(X_{n\wedge \tau })$ is itself a martingale — this is the discrete stochastic integral ${\sum }_k{H}_k(X_k-X_{k-1})$ with the predictable strategy $H_k=1_{\{\tau \ge k\}}$ — so $E[X_{n\wedge \tau }]=E[X_0]$ for every $n$, and the conditions license passing to the limit. $\square$

"You cannot beat a fair game." The martingale property is preserved by every predictable strategy: ${\sum }_k{H}_k(X_k-X_{k-1})$ has mean zero whenever $H$ is predictable and suitably integrable. This is a theorem, not a slogan, and the hypotheses are where the classical "martingale" betting system lives: doubling after each loss gives $\tau <\infty$ a.s. and $X_{\tau }=1$ a.s., apparently contradicting $E[X_{\tau }]=0$. What fails is condition (2) — the stopped process is not uniformly integrable, since the required capital is unbounded. Unlimited credit, not clever betting, is what breaks the theorem.

3. Convergence

Doob's upcrossing inequality. If $(X_n)$ is a supermartingale and $U_N[a,b]$ counts upcrossings of $[a,b]$ up to time $N$, then $(b-a)E[U_N[a,b]]\le E[(X_N-a{)}^{-}]$.

Martingale convergence theorem. If $(X_n)$ is a supermartingale bounded in $L^1$ (${\sup }_{n}E|X_n|<\infty$), then $X_n\to X_{\infty }$ almost surely with $X_{\infty }\in L^1$.

Proof. Failure of convergence means $\mathrm{liminf}<a<b<\mathrm{limsup}$ for some rationals $a<b$, which forces infinitely many upcrossings; the inequality bounds their expected number, so each such event is null, and there are countably many pairs $(a,b)$. $\square$

The proof is a good example of the general strategy: a deterministic criterion for convergence (finitely many upcrossings of every rational interval), converted into an almost-sure statement by an expectation bound plus countability. No independence is used anywhere.

$L^1$-boundedness does not give convergence in $L^1$: the martingale $X_n={\prod }_{i\le n}(1+{\epsilon }_i)$ with i.i.d. $\pm 1$ signs ${\epsilon }_i$ has $E[X_n]=1$ but $X_n\to 0$ a.s. Mass escapes, exactly as in convergence-modes.md §3. The repair is uniform integrability:

Theorem ($L^1$ convergence and closure). For a martingale $(X_n)$ the following are equivalent: (i) $(X_n)$ is UI; (ii) $X_n$ converges a.s. and in $L^1$; (iii) there is $Y\in L^1$ with $X_n=E[Y\mid {\mathcal{F}}_n]$ for all $n$ (the martingale is closed). In that case $X_{\infty }=E[Y\mid >{\mathcal{F}}_{\infty }]$.

Lévy's upward theorem. For $Y\in L^1$ and ${\mathcal{F}}_n\uparrow >{\mathcal{F}}_{\infty }$, $$E[Y\mid {\mathcal{F}}_n]⟶E[Y\mid {\mathcal{F}}_{\infty }]\text{a.s. and in }{L}^1.$$

Taking $Y=1_A$ with $A\in {\mathcal{F}}_{\infty }$ gives Lévy's zero–one law §4. There is a matching downward theorem for decreasing filtrations, and it is the standard route to de Finetti's theorem (exchangeability-de-finetti.md) and to Etemadi-style proofs of the strong law.

Under $L^p$ boundedness for $p>1$ everything is easier: Doob's maximal inequality $\Vert {\sup }_{k\le n}|X_k|{\Vert }_p\le \frac{p}{p-1}\Vert X_n{\Vert }_p$ gives $L^p$ boundedness of the maximum, hence UI, hence convergence in $L^p$.

4. Why Martingales Belong in a Foundations Folder

• They generalize the limit theorems. Part B's results assume independence; the martingale convergence theorem assumes only a conditional-mean structure, and martingale versions of the CLT and the LLN follow. Independence is revealed as sufficient rather than necessary.
• They make sequential conditioning well posed. The Borel–Kolmogorov ambiguity (conditional-expectation.md §6) arises when a null event is approached by an unspecified limiting procedure. A filtration specifies the procedure in advance, and Lévy's theorem then says the conditional probabilities converge to the truth. The lesson is that conditioning needs a conditioning structure, not that conditioning is defective.
• They formalize "information". Adaptedness, predictability, and stopping times give precise content to knowing, deciding in advance, and observing — notions that the informal vocabulary of probability uses constantly and cannot otherwise define.
• Optional stopping is a genuine impossibility theorem. It says no predictable strategy converts a fair game into a favourable one, with explicitly stated hypotheses that identify exactly which idealization ("unbounded capital") the familiar counterexamples exploit.

5. Where This Is Used

• Zero–one laws §4 — Lévy's law.
• Exchangeability and de Finetti — the downward theorem.
• Brownian motion — the continuous-time analogues; $B_t$ and $B_t^2-t$ are martingales, and the Itô integral is the continuous stochastic integral of §2.
• Conditional expectation — the tower property is the whole subject.