Σ-Algebra of τ-past

The σ-algebra of τ-past, (also named stopped σ-algebra, stopped σ-field, or σ-algebra 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

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

is called the σ-algebra of τ-past.

Properties

Monotonicity

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

$\sigma \leq \tau$

almost surely, then

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

Measurability

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