σ-Algebra of τ-past

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.^[1][2]

Definition[]

Let ${\displaystyle \tau }$ be a stopping time on the filtered probability space ${\displaystyle (\Omega ,{\mathcal {A}},({\mathcal {F}}_{t})_{t\in T},P)}$. Then the σ-algebra

${\displaystyle {\mathcal {F}}_{\tau }:=\{A\in {\mathcal {A}}\mid \{\tau \leq t\}\cap A\in {\mathcal {F}}_{t}{\text{ for all }}t\in T\}}$

is called the σ-algebra of τ-past.^[1][2]

Properties[]

Monotonicity[]

Is ${\displaystyle \sigma ,\tau }$ are two stopping times and

${\displaystyle \sigma \leq \tau }$

almost surely, then

${\displaystyle {\mathcal {F}}_{\sigma }\subset {\mathcal {F}}_{\tau }.}$

Measurability[]

A stopping time ${\displaystyle \tau }$ is always ${\displaystyle {\mathcal {F}}_{\tau }}$-measurable

Intuition[]

The same way ${\displaystyle F_{t}}$ is all the information up to time ${\displaystyle t}$, ${\displaystyle F_{\tau }}$ is all the information up time ${\displaystyle \tau }$. The only difference is that ${\displaystyle \tau }$ 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 ${\displaystyle \tau }$ be the first time the random walk hit 10, ${\displaystyle F_{\tau }}$ would give you the information to answer that question.^[3]

References[]