Σ-Algebra of τ-past

Template:Lowercase Template:Short description 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 $(\mathrm{\Omega },𝒜,({ℱ}_t{)}_{t\in T},P)$. Then the σ-algebra

${ℱ}_{\tau }:=\{A\in 𝒜\mid \mathrm{\forall }t\in T:\{\tau \le t\}\cap A\in {ℱ}_t\}$

is called the σ-algebra of τ-past.

Properties

Monotonicity

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

$\sigma \le \tau$

almost surely, then

${ℱ}_{\sigma }\subset {ℱ}_{\tau }.$

Measurability

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

Intuition

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