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

References

1. Karandikar, Rajeeva (2018). Introduction to Stochastic Calculus. Singapore: Springer Nature. p. 47. doi:10.1007/978-981-10-8318-1. ISBN 978-981-10-8317-4.
2. Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 193. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.