$\sigma$-algebra at a stopping time

Let ${({ℱ}_t)}_{t\in 𝕋}$ be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space $(\mathrm{\Omega },ℱ)$. For every $t\in 𝕋$, the $\sigma$-algebra ${ℱ}_t$ represents the collection of events which are observable up until time $t$. This concept can be generalized to any stopping time $\tau :\mathrm{\Omega }\to 𝕋\cup \{\mathrm{\infty }\}$.

For a stopping time $\tau$, the collection of events observable up until time $\tau$ is denoted by ${ℱ}_{\tau }$ and is generated by sampling progressively measurable processes

$${ℱ}_{\tau }=\sigma \left(\{X_{\tau \wedge t}:X\text{ is progressive, }t\in 𝕋\}\right).$$

The reason for sampling $X$ at time $\tau \wedge t$ rather than at $\tau$ is to include the possibility that $\tau =\mathrm{\infty }$, in which case $X_{\tau }$ is not defined.

A random variable $V$ is ${ℱ}_{\tau }$-measurable if and only if it is ${ℱ}_{\mathrm{\infty }}$-measurable and the process $X_t\equiv 1_{\{\tau \le t\}}V$ is adapted.

This can be shown as follows. If $X$ is a progressively measurable process, then the stopped process $X^{\tau \wedge s}$ is also progressive. In particular, $V\equiv X_{\tau \wedge s}=X_s^{\tau \wedge s}$ is ${ℱ}_{\mathrm{\infty }}$-measurable and $1_{\{\tau \le t\}}V=1_{\{\tau \le t\}}{X}_t^{\tau \wedge s}$ is ${ℱ}_t$-measurable.Conversely, if $V$ is ${ℱ}_t$-measurable then $X_s\equiv 1_{\{s>t\}}V$ is a progressive process and $1_{\{\tau >t\}}V=X_{\tau \wedge t}$ is ${ℱ}_{\tau }$-measurable. By letting $t$ increase to infinity, it follows that $1_{\{\tau =\mathrm{\infty }\}}V$ is ${ℱ}_{\tau }$-measurable for every ${ℱ}_{\mathrm{\infty }}$-measurable random variable $V$.Now suppose also that $X_t\equiv 1_{\{\tau \le t\}}V$ is adapted, and hence progressive. Then, $1_{\{\tau \le t\}}V=X_{\tau \wedge t}$ is ${ℱ}_{\tau }$-measurable. Letting $t$ increase to infinity shows that is ${ℱ}_{\tau }$-measurable.

As a set $A$ is ${ℱ}_{\tau }$-measurable if and only if $1_A$ is an ${ℱ}_{\tau }$-measurable random variable, this gives the following alternative definition,

$${ℱ}_{\tau }=\{A\in {ℱ}_{\mathrm{\infty }}:A\cap \{\tau \le t\}\in {ℱ}_t\text{ for all }t\in 𝕋\}.$$

From this, it is not difficult to show that the following properties are satisfied

1. Any stopping time $\tau$ is ${ℱ}_{\tau }$-measurable.

2. If $\tau (\omega )=t\in 𝕋\cup \{\mathrm{\infty }\}$ for all $\omega \in \mathrm{\Omega }$ then ${ℱ}_{\tau }={ℱ}_t$.

3. If $\sigma ,\tau$ are stopping times and $A\in {ℱ}_{\sigma }$ then $A\cap \{\sigma \le \tau \}\in {ℱ}_{\tau }$. In particular, if $\sigma \le \tau$ then ${ℱ}_{\sigma }\subseteq {ℱ}_{\tau }$.

4. If $\sigma ,\tau$ are stopping times and $A\in {ℱ}_{\sigma }$ then $A\cap \{\sigma =\tau \}\in {ℱ}_{\tau }$.

5. if the filtration $({ℱ}_t)$ is right-continuous and ${\tau }_n\ge \tau$ are stopping times with ${\tau }_n\to \tau$ then ${ℱ}_{\tau }={\bigcap }_n{ℱ}_{{\tau }_n}$. More generally, if ${\tau }_n=\tau$ eventually then this is true irrespective of whether the filtration is right-continuous.

6. If ${\tau }_n$ are stopping times with ${\tau }_n=\tau$ eventually then ${ℱ}_{{\tau }_n}\to {ℱ}_{\tau }$. That is,

   $${ℱ}_{\tau }=\bigcap _n\sigma \left(\bigcup _{m\ge n}{ℱ}_{{\tau }_m}\right).$$

In continuous-time, for any stopping time $\tau$ the $\sigma$-algebra ${ℱ}_{\tau +}$ is the set of events observable up until time $t$ with respect to the right-continuous filtration $({ℱ}_{t+})$. That is,

If ${\tau }_n\ge \tau$ are stopping times with ${\tau }_n>\tau$ whenever is not a maximal element of $𝕋$, and ${\tau }_n\to \tau$ then,

$${ℱ}_{\tau +}=\bigcap _n{ℱ}_{{\tau }_n}=\bigcap _n{ℱ}_{{\tau }_n+}.$$

The $\sigma$-algebra of events observable up until just before time $\tau$ is denoted by ${ℱ}_{\tau -}$ and is generated by sampling predictable processes

$${ℱ}_{\tau -}=\sigma \left(\{X_{\tau \wedge t}:X\text{ is predictable, }t\in 𝕋\}\right).$$

Suppose that the index set $𝕋\subseteq ℝ$ has minimal element $t_0$.As the predictable $\sigma$-algebra is generated by sets of the form $(s,\mathrm{\infty })\times A$ for $s\in 𝕋$ and $A\in {ℱ}_s$, and $\{t_0\}\times A$ for $A\in {ℱ}_{t_0}$, the definition above can be rewritten as,

$${ℱ}_{\tau -}=\sigma (\{A\cap \{\tau >s\}:s\in 𝕋,A\in {ℱ}_s\}\cup {ℱ}_{t_0}).$$

Clearly, ${ℱ}_{\tau -}\subseteq {ℱ}_{\tau }\subseteq {ℱ}_{\tau +}$. Furthermore, for any stopping times $\sigma ,\tau$ then ${ℱ}_{\sigma +}\subseteq {ℱ}_{\tau -}$ when restricted to the set .

If ${\tau }_n$ is a sequence of stopping times announcing (http://planetmath.org/PredictableStoppingTime) $\tau$, so that $\tau$ is predictable, then

$${ℱ}_{\tau -}=\sigma \left(\bigcup _n{ℱ}_{{\tau }_n}\right).$$