a predictable (previsible) process
Imagine you are gambling and you may change how much you stake from round to round. There is one ironclad rule: you must decide each round's bet BEFORE that round is played, using only what you have already seen. You cannot size your bet on round n by peeking at round n's result — that would be cheating. A process that obeys exactly this rule, where the value at time n is fixed using only the information available at the end of time n-1, is called predictable (the British literature says previsible). It is the formal model of an honest, non-clairvoyant betting strategy.
Precisely, a process H_1, H_2, H_3, ... is predictable with respect to a filtration if each H_n is F_(n-1)-measurable. Compare this with 'adapted', the condition a martingale satisfies, where M_n need only be F_n-measurable. Predictable is one notch stronger and one step earlier: H_n must be knowable a full step before time n, with no information from F_n leaking in. The name fits — H_n is 'predictable' in the sense that it is already determined before the n-th increment happens; it holds no surprise of its own. A constant strategy and 'bet last round's outcome' are both predictable; 'bet this round's outcome' is not.
Predictable processes are the inputs to the martingale transform, the discrete analogue of an integrand in a stochastic integral. The whole reason gambling systems cannot beat a fair game rests on demanding the stake be predictable: if you were allowed to set your bet knowing the outcome, you could obviously win, so the no-clairvoyance requirement is exactly what makes 'you cannot beat a fair game' a theorem rather than a falsehood. In continuous time the same idea becomes the technical heart of the Ito integral, where the integrand must be predictable so that you are never integrating against the future.
On a fair coin game let H_n be your stake on round n. 'Always bet 2 dollars' and 'double last round's stake after a loss' (the classic martingale strategy) are both predictable: H_n uses only rounds 1 through n-1. But 'bet 100 dollars exactly when round n comes up heads' is NOT predictable — it requires knowing round n's outcome before betting, which no honest strategy can.
Predictable = decided one step ahead, using only the past; this is what makes a betting strategy honest.
Predictable (H_n is F_(n-1)-measurable) is strictly stronger than adapted (F_n-measurable): the value must be known one full step before its increment, with no peek at the present.