Doob's martingale

A Doob’s martingale is a mathematical model describing a fair game or process where the best prediction for the next outcome, given all previous information, is simply the most recent outcome. Think of it like a fair betting game: your expected future winnings, based on what's happened so far, is always equal to your current winnings. Martingales are useful in fields like finance and probability theory because they capture the idea of no predictable trend or bias in the process, ensuring that, on average, you neither gain nor lose over time when considering all past information.