Σ-Algebra of τ-past explained

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.

Definition

Let

\tau be a stopping time on the filtered probability space (\Omega, l A, ( l F _{t)} _{t} , P ) . Then the σ-algebral F _{\tau:=} \{ A \in l A \mid \forall t \in T \colon \{ \tau \leq t \} \cap A \in l F _{t\}}

is called the σ-algebra of τ-past.

Properties

Monotonicity

Is

\sigma, \tau are two stopping times and\sigma \leq \tau almost surely, thenl F _{\sigma} \subset l F _{\tau.}

Measurability

A stopping time

\tau is always l F _{\tau} -measurable.

Intuition

The same way

l{F} _{t} is all the information up to time t , l{F} _{\tau} is all the information up time \tau . The only difference is that \tau 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 \tau be the first time the random walk hit 10, l{F} _{\tau} would give you the information to answer that question.

References

^{[1] [2] [3]}

Notes and References

1. Book: Karandikar . Rajeeva . 2018 . Introduction to Stochastic Calculus . Indian Statistical Institute Series . Singapore . Springer Nature. 10.1007/978-981-10-8318-1 . 978-981-10-8317-4. 47 .
2. Book: Klenke . Achim . 2008 . Probability Theory . Berlin . Springer . 10.1007/978-1-84800-048-3 . 978-1-84800-047-6. 193 .
3. Web site: Earnest, Mike (2017). Comment on StackExchange: Intuition regarding the σ algebra of the past (stopping times).