Filtration (probability theory)

A filtration is an increasing family of sigma-algebras that models information available over time. Each later sigma-algebra contains everything measurable from earlier ones.

Sigma-algebra: A collection of sets closed under complements and countable unions, used to define measurable events. Each stage of a filtration is a sigma-algebra of events known by that time.
Probability space: A triple consisting of a sample space, a sigma-algebra of events, and a probability measure. Every filtration is defined on the sigma-algebra of an underlying probability space.
Stochastic process: A collection of random variables indexed by time or another parameter. A filtration specifies the information available alongside a stochastic process.
Augmented filtration: A filtration enlarged by null events and made right-continuous under standard probability-space assumptions. It modifies a given filtration to meet regularity conditions used in stochastic calculus.
Adapted process: A stochastic process whose value at each time is measurable with respect to the information available then. Adaptation expresses that a process uses no information beyond the filtration at each time.
Measurable function: A function whose inverse images of measurable sets are measurable. Measurability determines whether a random variable is observable at a filtration stage.
Stochastic calculus: Calculus for stochastic processes, including integration with respect to random processes. Its integrands are generally required to respect the information represented by a filtration.
Usual conditions: The requirements that a filtration be complete and right-continuous. These conditions refine the raw information sequence without changing its basic role.
Stopping time: A random time whose occurrence by any fixed time can be determined from information available by then. Its definition tests whether the filtration captures information progressively.
Random variable: A measurable function from a probability space to a value space. Events about a random variable belong to a stage when its value is known there.

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