Martingales — Mathematics Knowledge System

Overview

A martingale is a stochastic process whose future expected value, given all past information, equals its current value. Martingales model fair games: knowing the full history of a fair coin game gives no advantage for predicting future net winnings. Doob's theory of martingales provides powerful convergence theorems and inequalities that underpin modern probability, stochastic calculus, and mathematical finance.

Intuition

A martingale is a model of a fair game. If {X_n} tracks your wealth in a fair casino, then no matter how long you have played and what happened, your expected future wealth equals your current wealth. Geometrically, the martingale property says the process has no 'drift' — conditional expectations project forward without bias.

Formal Definition

Definition

Let (Ω, F, P) be a probability space with filtration {F_n}_{n≥0} (an increasing sequence of sub-σ-algebras). A sequence {X_n} is a martingale with respect to {F_n} if:

X_n \in L^1(P) \text{ and } X_n \text{ is } \mathcal{F}_n\text{-measurable for each } n

Adaptedness and integrability

MG-1

E[X_{n+1} \mid \mathcal{F}_n] = X_n \quad \text{a.s., for all } n \geq 0

Martingale property

MG-2

Notation

Notation	Meaning
$\{\mathcal{F}_n\}_{n \geq 0}$	Filtration: increasing sequence of σ-algebras representing information up to time n
$(X_n, \mathcal{F}_n)_{n\geq 0}$	Martingale adapted to the filtration
$X^*_n = \max_{0 \leq k \leq n} X_k$	Running maximum of the martingale

Properties

Submartingale

E[X_{n+1} \mid \mathcal{F}_n] \geq X_n

Condition: Models a game favourable to the player; convex functions of martingales are submartingales by Jensen

Supermartingale

E[X_{n+1} \mid \mathcal{F}_n] \leq X_n

Condition: Models a game unfavourable to the player

Optional stopping theorem

E[X_\tau] = E[X_0] \text{ for bounded stopping times } \tau

Condition: T bounded or {X_{n∧τ}} uniformly integrable

Doob's maximal inequality

P\!\left(\max_{0\leq k\leq n} X_k \geq \lambda\right) \leq \frac{E[X_n^+]}{\lambda}

Condition: X submartingale, λ > 0

Doob's L^p inequality

\|X^*_n\|_p \leq \frac{p}{p-1}\|X_n\|_p

Condition: X non-negative submartingale, p > 1

Theorems

Theorem 1: Doob's Martingale Convergence Theorem

\text{If } \{X_n\} \text{ is a martingale with } \sup_n E[|X_n|] < \infty, \text{ then } X_\infty = \lim_{n\to\infty} X_n \text{ exists a.s.}

Theorem 2: Doob's Optional Stopping Theorem

\text{If } \tau \text{ is a stopping time and } \{X_{n \wedge \tau}\} \text{ is uniformly integrable, then } E[X_\tau] = E[X_0]

Theorem 3: Doob Decomposition

\text{Any adapted integrable process } X_n = M_n + A_n \text{ where } M_n \text{ is a martingale and } A_n \text{ is predictable and non-decreasing (unique a.s.)}

Theorem 4: Lévy's Upward Theorem

\text{If } X \in L^1 \text{ and } \mathcal{F}_n \nearrow \mathcal{F}_\infty, \text{ then } E[X \mid \mathcal{F}_n] \to E[X \mid \mathcal{F}_\infty] \text{ a.s. and in } L^1

Applications

The fundamental theorem of asset pricing: a market is arbitrage-free iff there exists an equivalent martingale measure under which discounted asset prices are martingales.

Worked Examples

Martingale property: E[S_{n+1}|F_n] = S_n + E[X_{n+1}|F_n] = S_n + E[X_{n+1}] = S_n + 0 = S_n.
$E[S_{n+1} \mid \mathcal{F}_n] = S_n$
τ is a bounded stopping time (actually finite a.s. by recurrence). By the optional stopping theorem:
$E[S_\tau] = E[S_0] = 0$
S_τ takes values a (with probability p) and -b (with probability 1-p), so:
$pa + (1-p)(-b) = 0 \implies p = \frac{b}{a+b}$

Answer: P(reaching a before -b) = b/(a+b), confirming the classical gambler's ruin formula.

Practice Problems

Difficulty 7/10

Prove that if {M_n} is a martingale and φ is convex, then {φ(M_n)} is a submartingale (assuming integrability).

Difficulty 8/10

A gambler plays a fair game starting with $k, stopping at $0 (ruin) or $N (success). Use martingale theory to find the probability of reaching $N before ruin.

Difficulty 9/10

State and prove Doob's upcrossing inequality, and deduce the martingale convergence theorem.

Common Mistakes

Common Mistake

Any bounded stopping time can be used in optional stopping without further conditions

Optional stopping requires either τ bounded or {X_{n∧τ}} uniformly integrable. Counterexample: doubling strategy in a fair game with τ = first win reaches E[S_τ] = 1 ≠ 0 = E[S_0] when τ is only a.s. finite but not bounded.

Common Mistake

A martingale converges in L¹ whenever it converges a.s.

A.s. convergence does not imply L¹ convergence without uniform integrability. The process can converge a.s. to 0 while E[|X_n|] → ∞.

Historical Background

The term 'martingale' originally referred to a betting strategy of doubling stakes after each loss. Lévy (1935) introduced martingales in the context of conditional expectations, and Doob (1940–1953) developed the systematic theory including optional stopping, convergence theorems, and maximal inequalities. Martingales became central to mathematical finance through the work of Harrison, Kreps, and Pliska (1979–1981).

1935
Lévy introduces conditional expectations as a proto-martingale notion
Paul Lévy
1940
Doob develops martingale theory and the optional stopping theorem
Joseph Doob
1953
Doob's monograph Stochastic Processes systematises the theory
Joseph Doob
1979
Harrison–Kreps: equivalent martingale measures and no-arbitrage in finance
Michael Harrison, David Kreps

Summary

A martingale satisfies E[X_{n+1}|F_n] = X_n a.s., modelling a fair game.
Doob's maximal and L^p inequalities control the running maximum of a martingale.
Convergence theorem: L¹-bounded martingales converge a.s.
Optional stopping: E[X_τ] = E[X_0] under uniform integrability of {X_{n∧τ}}.
Martingales are the foundation of risk-neutral pricing in mathematical finance.

References

BookWilliams, D. (1991). Probability with Martingales. Cambridge University Press.
BookDurrett, R. (2019). Probability: Theory and Examples (5th ed.). Cambridge University Press.