A martingale is a stochastic process that represents a fair game, i.e. the conditional expectation of the next value given all past values = the present value.
If for all ,
then is a martingale.
Intuition
At any point, the expected future value is just the present value—nothing more, nothing less. The current position is all you need because future changes are independent of the past path that led to the current state.
The past holds no predictive power.
Stopping a martingale does not change the martingale’s outcome
It’s easy to think that by stopping at a specific time (especially at a “lucky” time), one might gain an advantage. For instance, in gambling, people often believe they can “cash out” at a good time to ensure profits (Gambler’s fallacy).
But stopping a martingale at a stopping time doesn’t change the fact that it’s still a martingale.
No stopping rule, no matter how cleverly devised, can create a winning strategy in a fair game.