Martingales

A martingale is a stochastic process that models a "fair game": the expected future value, given all past information, equals the present value. Martingales are central to modern probability theory, providing a unified framework for analyzing random walks, gambling strategies, financial derivatives, and stochastic integrals.

Definition

Definition3.1Martingale

A stochastic process $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal{F}_n)$ is a martingale if:

$M_n$ is integrable: $\mathbb{E}[|M_n|] < \infty$ for all $n$ .
Martingale property: For all $n \geq 0$ ,

$\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = M_n \quad \text{almost surely}.$

A process is a submartingale if $\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] \geq M_n$ and a supermartingale if $\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] \leq M_n$ .

Intuition: In a martingale, the "best guess" for tomorrow's value, given all information up to today, is today's value. There is no systematic upward or downward trend.

RemarkFiltration

A filtration $(\mathcal{F}_n)$ is an increasing sequence of $\sigma$ -algebras representing "information available up to time $n$ ." The process $(M_n)$ is adapted if $M_n$ is $\mathcal{F}_n$ -measurable for all $n$ (i.e., $M_n$ depends only on information up to time $n$ ).

Often, we take $\mathcal{F}_n = \sigma(M_0, M_1, \ldots, M_n)$ , the natural filtration generated by the process itself.

Basic examples

ExampleSimple random walk

Let $X_0 = 0$ and $X_n = \xi_1 + \cdots + \xi_n$ , where $\xi_i$ are i.i.d. with $\mathbb{P}(\xi_i = 1) = \mathbb{P}(\xi_i = -1) = 1/2$ . Then:

$\mathbb{E}[X_{n+1} \mid X_0, \ldots, X_n] = X_n + \mathbb{E}[\xi_{n+1}] = X_n.$

So $(X_n)$ is a martingale. The symmetric random walk is the prototypical discrete-time martingale.

ExampleSquared random walk minus time

Let $X_n$ be the simple random walk. Define $M_n = X_n^2 - n$ . Then:

$\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = \mathbb{E}[X_{n+1}^2 - (n+1) \mid \mathcal{F}_n] = \mathbb{E}[(X_n + \xi_{n+1})^2 \mid \mathcal{F}_n] - n - 1.$

Since $\xi_{n+1}$ is independent of $\mathcal{F}_n$ and $\mathbb{E}[\xi_{n+1}] = 0$ , $\mathbb{E}[\xi_{n+1}^2] = 1$ :

$\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = X_n^2 + 1 - n - 1 = M_n.$

So $(M_n)$ is also a martingale. This is a "compensated" process: $X_n^2$ is a submartingale (growing on average), and subtracting its mean growth $n$ produces a martingale.

ExampleProduct of independent fair games

Let $\xi_1, \xi_2, \ldots$ be i.i.d. with $\mathbb{E}[\xi_i] = 1$ . Define $M_n = \prod_{i=1}^n \xi_i$ (with $M_0 = 1$ ). Then:

$\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = M_n \mathbb{E}[\xi_{n+1}] = M_n.$

So $(M_n)$ is a martingale. This models a multiplicative fair game, such as repeated betting with independent, fair odds.

Martingale transform (betting strategies)

Definition3.2Predictable process

A process $(C_n)$ is predictable with respect to $(\mathcal{F}_n)$ if $C_n$ is $\mathcal{F}_{n-1}$ -measurable for all $n \geq 1$ . Intuitively, $C_n$ is chosen based on information up to time $n-1$ , before observing $M_n$ .

Theorem3.1Martingale transform

If $(M_n)$ is a martingale and $(C_n)$ is a bounded predictable process, then the martingale transform

$(C \cdot M)_n = \sum_{k=1}^n C_k (M_k - M_{k-1})$

is also a martingale.

Interpretation: $C_n$ represents a "betting strategy" (how much to bet at time $n$ ), and $M_k - M_{k-1}$ is the "gain" at time $k$ . The martingale transform says that no betting strategy can convert a fair game into a favorable one — the resulting wealth process is still a martingale.

ExampleDoubling strategy fails

Consider a gambler playing a fair coin flip game, starting with wealth $M_0 = 1$ . The "doubling strategy" is: bet $C_n = 2^{n-1}$ (double the bet after each loss). If the first win occurs at time $N$ , the gambler's wealth is:

$M_N = 1 - (1 + 2 + \cdots + 2^{N-1}) + 2^{N} = 2.$

It seems the gambler always wins! However, $\mathbb{P}(N < \infty) = 1$ but $\mathbb{E}[N] = \infty$ . The strategy is not bounded (it requires unbounded capital), so the martingale transform theorem does not apply. In fact, if the gambler has finite capital, the strategy eventually fails with positive probability.

Optional stopping

Theorem3.2Optional stopping theorem (simple form)

Let $(M_n)$ be a martingale and $\tau$ a stopping time with $\mathbb{E}[\tau] < \infty$ and $\sup_n |M_n|$ integrable. Then:

$\mathbb{E}[M_\tau] = \mathbb{E}[M_0].$

This theorem says that, under appropriate conditions, the expected value of the martingale at a stopping time equals its initial value. "You can't beat the house by choosing when to stop."

ExampleGambler's ruin via optional stopping

Let $X_n$ be a simple random walk on $\{0, 1, \ldots, N\}$ with absorbing boundaries. Let $\tau = \inf\{n : X_n \in \{0, N\}\}$ be the exit time. Since $X_n$ is a martingale (before absorption) and $\tau$ is finite a.s., by optional stopping:

$\mathbb{E}[X_\tau \mid X_0 = k] = k.$

But $X_\tau \in \{0, N\}$ , so if $h_k = \mathbb{P}(X_\tau = N \mid X_0 = k)$ :

$k = 0 \cdot (1-h_k) + N \cdot h_k \Rightarrow h_k = k/N.$

This gives the probability of reaching $N$ before $0$ , starting from $k$ , for a symmetric random walk.

Doob decomposition

Theorem3.3Doob decomposition

Any adapted, integrable process $(X_n)$ admits a unique decomposition:

$X_n = M_n + A_n,$

where $(M_n)$ is a martingale with $M_0 = 0$ and $(A_n)$ is a predictable process with $A_0 = 0$ . The predictable part is:

$A_n = \sum_{k=1}^n \mathbb{E}[X_k - X_{k-1} \mid \mathcal{F}_{k-1}].$

This decomposition separates the "martingale part" (unpredictable fluctuations) from the "drift" (systematic trend). For a submartingale, $A_n$ is increasing; for a supermartingale, $A_n$ is decreasing.

ExampleAsymmetric random walk

Let $X_n = \sum_{i=1}^n \xi_i$ , where $\mathbb{P}(\xi_i = 1) = p$ , $\mathbb{P}(\xi_i = -1) = q = 1-p$ . Then $\mathbb{E}[\xi_i] = p - q = 2p - 1 \neq 0$ . The Doob decomposition is:

$X_n = M_n + (2p-1)n,$

where $M_n = X_n - (2p-1)n$ is a martingale (the "centered" walk) and $A_n = (2p-1)n$ is the drift.

Summary

Martingales capture the notion of a "fair" stochastic process:

Definition: $\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = M_n$ (conditional expectation preserves current value).
Martingale transform: No betting strategy can make a fair game favorable.
Optional stopping: Expected value at a stopping time equals initial value (under appropriate conditions).
Doob decomposition: Any process = martingale + predictable drift.

Martingales are the foundation for stochastic calculus, martingale convergence theorems, and the modern theory of stochastic integration.