Martingale

A martingale is an integrable stochastic process whose conditional expected future value, given all information available now, equals its present value. It models a fair game with no predictable gain.

Conditional expectation: A random variable’s expected value given specified information, represented by a random variable measurable with respect to that information. The martingale condition states that conditioning a future value on present information recovers the present value.
Stochastic process: A collection of random variables indexed by time or another parameter. A martingale is a stochastic process subject to an additional conditional-mean constraint.
Risk-neutral measure: A probability measure under which discounted asset prices are martingales in an arbitrage-free market model. Derivative pricing uses martingale expectations under this measure.
Submartingale: An integrable adapted process whose conditional expected future value is at least its present value. It permits nonnegative conditional drift where a martingale requires equality.
Filtration (probability theory): An increasing sequence of sigma-algebras representing the information available over time in a stochastic process. A martingale’s conditional expectations are taken with respect to this growing information.
Adapted process: A stochastic process whose value at each time is measurable with respect to the information available by that time. Martingales must not use information from the future when assigning present values.
Black–Scholes model: A continuous-time model for asset prices and options based on geometric Brownian motion and no-arbitrage assumptions. Its discounted stock price is a martingale under the risk-neutral measure.
Supermartingale: An integrable adapted process whose conditional expected future value is at most its present value. It describes conditional downward drift rather than the martingale’s fair-game balance.
Optional stopping theorem: A family of results giving conditions under which stopping a martingale at a random time preserves its expected value. It formalizes when the fair-game expectation survives a strategy that chooses when to stop.
Sigma-algebra: A collection of subsets of a sample space that contains the whole space and is closed under complements and countable unions. A filtration consists of sigma-algebras that encode the process’s available information.

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