Martingales

Fair games: given the history, tomorrow's expected value is exactly today's.

The idea

A martingale is the mathematical model of a fair game.

Suppose you play some gambling game round after round, and write $M_{n}$ for your fortune after $n$ rounds. What makes the game fair? Not that your fortune stays put — it rises and falls. Not that you break even in the end — a fair game can still leave you rich or ruined. The condition that captures fairness concerns the round ahead: whatever has happened so far, the next round must be worth nothing to you, in either direction. Conditioning on the whole past, this says

$\mathbb{E}[M_{n+1} \mid M_{0}, M_{1}, \dots, M_{n}] = M_{n},$

which we read as the expected value of $M_{n+1}$, given the history so far, is $M_{n}$. A process satisfying this at every $n$ is a martingale. The left side is the best forecast of tomorrow's fortune, made with everything known today; the right side is today's fortune itself. The definition constrains the values not at all — they may wander anywhere — but it constrains every increment: given the past, each one averages to zero. Pictured from round $n$: the possible futures fan out from the current fortune $M_{n}$, and however far they spread, their conditional average stays level at $M_{n}$.

Ways to work on it

• Walkthrough. The fair-game definition, examples built from random walks, and a non-example with drift.
• Practice. Test processes for the martingale property, and cancel drift to create new ones.
• Hardest. Verify the exponential martingale hidden inside a biased random walk.

Not sure where to start? Take the ten-question placement test.