Martingales and Optional Stopping
A martingale is a process whose expected next value, given everything so far, is its current value. Stopping it at a random time preserves that expectation, as long as the stopping rule is not allowed to look ahead.
What a learner can do afterwards
- Check the martingale property for a given process
- Apply optional stopping to compute a ruin probability
- Give a stopping rule for which the theorem's hypotheses fail and the conclusion is false
1 · Read
A martingale is a process whose expected next value, given everything so far, equals its current value. The symmetric walk with steps plus 1 or minus 1 is the model: each step has mean 0, so starting from 0 the mean after 3 steps is still 0. Drifted walks, squares, and absolute values all trend and fail the test.
Think of a fair coin game priced at zero: neither side profits on average, and repeating the game cannot change that without new information. The binomial coin flips are the laboratory where fairness is first measured. Only the symmetric walk keeps zero-mean increments given the past.
Optional stopping says you may stop a martingale at a random time and keep the same expectation, if the rule never looks ahead and stays bounded. A gambler aiming at a target fortune before ruin uses this: the stopped mean still equals the starting fortune, which unlocks the ruin probability. Boundedness, of time or fortune, is the price of the conclusion.
A rule that peeks at the future breaks the theorem, and the conclusion can fail badly. Waiting until the peak of a finished path, or doubling stakes without limit until a win, both look ahead or run unbounded. When you audit a stopping rule, check foresight first and boundedness second.
Martingales forecast tomorrow as today, fair stopping preserves the mean, and any rule with foresight or no bound voids the promise.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.