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.

Definition

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

${\mathcal {F}}_{\tau }:=\{A\in {\mathcal {A}}\mid \forall t\in T\colon \{\tau \leq t\}\cap A\in {\mathcal {F}}_{t}\}$

is called the σ-algebra of τ-past.

Properties

Monotonicity

If $\sigma ,\tau$ are two stopping times and

$\sigma \leq \tau$

almost surely, then

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

Measurability

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

Intuition

The same way ${\mathcal {F}}_{t}$ is all the information up to time $t$, ${\mathcal {F}}_{\tau }$ 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, ${\mathcal {F}}_{\tau }$ would give you the information to answer that question.