In probability theory, a filtration is an increasing family of sigma-algebras indexed by time that formally represents the accumulation of information available up to each instant. It models what is knowable at each point as a stochastic process unfolds, with each sigma-algebra containing all events whose outcomes are determined by then. Filtrations are fundamental to defining adapted processes, martingales, and conditional expectations in stochastic analysis and mathematical finance.
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- Formally, a filtration is a collection of nested sigma-algebras where earlier ones are contained in later ones, capturing the irreversible growth of knowledge over time. A process is adapted to a filtration when its value at each time is measurable with respect to the corresponding sigma-algebra, a condition essential to defining martingales, stopping times, and the conditional expectations central to sequential inference and stochastic modelling.