Martingale
A martingale is a process whose conditional expectation of the future, given the present, is the present — ‘fair game’ under that information and that measure.
Definition
E[X_{t+1} | F_t] = X_t. Sub- and super-martingales drift up or down. Asset prices discounted at the risk-free rate are martingales under a risk-neutral measure if the market is arbitrage-free (FTAP).
Why it matters
This is the sentence behind ‘the discounted asset is a martingale.’ Changing measure (Girsanov) is how drifts become r.
Case
A physical stock is not a martingale (it has a risk premium). The risk-neutral stock, discounted, is. Mixing the two measures is a common homework error and a common slide error.
How to read it
Name the filtration and the measure. A martingale for the street’s information is not a martingale for an insider.