σ-Algebra of τ-past

Short description: Algebra of a branch of probability theory

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 [math]\displaystyle{ \tau }[/math] be a stopping time on the filtered probability space [math]\displaystyle{ (\Omega, \mathcal A, (\mathcal F_t)_{t \in T}, P ) }[/math]. Then the σ-algebra

[math]\displaystyle{ \mathcal F_\tau:= \{ A \in \mathcal A \mid \forall t \in T \colon \{ \tau \leq t \} \cap A \in \mathcal F_t\} }[/math]

is called the σ-algebra of τ-past.

Properties

Monotonicity

Is [math]\displaystyle{ \sigma, \tau }[/math] are two stopping times and

[math]\displaystyle{ \sigma \leq \tau }[/math]

almost surely, then

[math]\displaystyle{ \mathcal F_\sigma \subset \mathcal F_\tau. }[/math]

Measurability

A stopping time [math]\displaystyle{ \tau }[/math] is always [math]\displaystyle{ \mathcal F_\tau }[/math]-measurable.

Intuition

The same way [math]\displaystyle{ \mathcal{F}_t }[/math] is all the information up to time [math]\displaystyle{ t }[/math], [math]\displaystyle{ \mathcal{F}_\tau }[/math] is all the information up time [math]\displaystyle{ \tau }[/math]. The only difference is that [math]\displaystyle{ \tau }[/math] 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 [math]\displaystyle{ \tau }[/math] be the first time the random walk hit 10, [math]\displaystyle{ \mathcal{F}_\tau }[/math] would give you the information to answer that question.