Optional stopping theorem

The optional stopping theorem gives conditions under which a martingale stopped at a random time has the same expected value as the initial martingale.

Martingale: A stochastic process whose conditional expected future value, given its history, equals its current value. The theorem applies to martingales and preserves their expected value under suitable stopping conditions.
St. Petersburg paradox: A probability paradox in which a game has infinite expected payout but a finite willingness to pay is commonly observed. Its unbounded payouts illustrate why expectation arguments can fail without adequate integrability.
Sequential probability ratio test: A sequential hypothesis test that accumulates likelihood evidence until it crosses a decision threshold. Its random sample size makes martingale stopping arguments relevant to error probabilities.
Conditional expectation: The expected value of a random variable given specified information. Martingale definitions and stopping arguments are expressed through conditional expectations.
Stopping time: A random time whose occurrence by any given moment can be determined from information available by then. The theorem concerns martingales evaluated at times defined without access to future information.
Gambler's ruin: A stochastic model of a gambler whose finite capital is lost or reaches a target through repeated random bets. Stopping at ruin or a target provides familiar settings where optional-stopping conditions must be checked.
Doob martingale: A martingale formed from conditional expectations of a random variable given an increasing sequence of information. Stopping such conditional-expectation processes can justify bounds in sequential analyses.
Filtration (probability theory): An increasing sequence of information sets describing what is known over time in a stochastic process. Stopping times and martingales are defined relative to evolving information.
Uniform integrability: A tail-control condition on a family of random variables that prevents expected values from escaping into rare, large outcomes. It supplies a powerful condition under which stopped martingales retain their expectations.
Doob's inequality: An inequality bounding the probability that a martingale’s maximum exceeds a specified level. It controls stopped-process excursions rather than asserting equality of expected values.

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Doob's martingale inequalityRelated: Both results use stopping times, but optional stopping concerns expectations at random times rather than maxima.