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 τ {\displaystyle \tau } be a stopping time on the filtered probability space ( Ω , A , ( F t ) t ∈ T , P ) {\displaystyle (\Omega ,{\mathcal {A}},({\mathcal {F}}_{t})_{t\in T},P)} . Then the σ-algebra
F τ := { A ∈ A ∣ ∀ t ∈ T : { τ ≤ t } ∩ A ∈ F t } {\displaystyle {\mathcal {F}}_{\tau }:=\{A\in {\mathcal {A}}\mid \forall t\in T\colon \{\tau \leq t\}\cap A\in {\mathcal {F}}_{t}\}}
is called the σ-algebra of τ-past.
Properties
Monotonicity If σ , τ {\displaystyle \sigma ,\tau } are two stopping times and
σ ≤ τ {\displaystyle \sigma \leq \tau }
almost surely, then
F σ ⊂ F τ . {\displaystyle {\mathcal {F}}_{\sigma }\subset {\mathcal {F}}_{\tau }.}
Measurability A stopping time τ {\displaystyle \tau } is always F τ {\displaystyle {\mathcal {F}}_{\tau }} -measurable.
Intuition The same way F t {\displaystyle {\mathcal {F}}_{t}} is all the information up to time t {\displaystyle t} , F τ {\displaystyle {\mathcal {F}}_{\tau }} is all the information up to time τ {\displaystyle \tau } . The only difference is that τ {\displaystyle \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 τ {\displaystyle \tau } be the first time the random walk hit 10, F τ {\displaystyle {\mathcal {F}}_{\tau }} would give you the information to answer that question.
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