Σ-Algebra of τ-pastThe σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-field of τ-past) is a σ-algebra associated with a stopping time in the theory of stochastic processes, a branch of probability theory.[1][2] DefinitionLet be a stopping time on the filtered probability space . Then the σ-algebra is called the σ-algebra of τ-past.[1][2] PropertiesMonotonicityIf are two stopping times and almost surely, then MeasurabilityA stopping time is always -measurable. IntuitionThe same way is all the information up to time , is all the information up time . The only difference is that is random. For example, if you had a random walk, and you wanted to ask, “How many times did the random walk hit −5 before it first hit 10?”, then letting be the first time the random walk hit 10, would give you the information to answer that question.[3] References
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