The 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.
be a stopping time on the filtered probability space
Then the σ-algebra is called the σ-algebra of τ-past.
σ , τ
are two stopping times and almost surely, then A stopping time
is all the information up to time
is all the information up time
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.