Martingales

A martingale is a stochastic process $X_t$ where the conditional expectation of the future given the past equals the present:

Translation: a martingale is a fair game. No matter what's happened so far, your expected future value is exactly where you are now.

Brownian motion is a martingale. So is a fair-coin gambling balance with one-unit wagers. Discounted asset prices under the risk-neutral measure are martingales — that's what risk-neutral valuation means.

The Optional Stopping Theorem says that for a "nice" stopping time $\tau$, the expectation at the stopping time equals the starting value: $E[X_\tau] = X_0$. This sounds obvious but yields powerful results — the "double-or-nothing" gambler's ruin, the expected hitting time of barriers, and many option-pricing identities.

The trap: optional stopping can fail when $\tau$ is unbounded and the martingale isn't uniformly integrable. The classic example is doubling your bet until you win — the strategy looks profitable but requires unlimited capital.