$\sigma$ -algebra at a stopping time

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

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

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

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

A random variable $V$ is $\mathcal{F}_{\tau}$ -measurable if and only if it is $\mathcal{F}_{\infty}$ -measurable and the process $X_{t}\equiv 1_{\{\tau\leq t\}}V$ 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, $V\equiv X_{\tau\wedge s}=X^{\tau\wedge s}_{s}$ is $\mathcal{F}_{\infty}$ -measurable and $1_{\{\tau\leq t\}}V=1_{\{\tau\leq t\}}X^{\tau\wedge s}_{t}$ is $\mathcal{F}_{t}$ -measurable. Conversely, if $V$ is $\mathcal{F}_{t}$ -measurable then $X_{s}\equiv 1_{\{s>t\}}V$ is a progressive process and $1_{\{\tau>t\}}V=X_{\tau\wedge t}$ is $\mathcal{F}_{\tau}$ -measurable. By letting $t$ increase to infinity, it follows that $1_{\{\tau=\infty\}}V$ is $\mathcal{F}_{\tau}$ -measurable for every $\mathcal{F}_{\infty}$ -measurable random variable $V$ . Now suppose also that $X_{t}\equiv 1_{\{\tau\leq t\}}V$ is adapted, and hence progressive. Then, $1_{\{\tau\leq t\}}V=X_{\tau\wedge t}$ is $\mathcal{F}_{\tau}$ -measurable. Letting $t$ increase to infinity shows that $V=1_{\{\tau<\infty\}}V+1_{\{\tau=\infty\}}V$ is $\mathcal{F}_{\tau}$ -measurable.

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

$\mathcal{F}_{\tau}=\left\{A\in\mathcal{F}_{\infty}:A\cap\{\tau\leq t\}\in% \mathcal{F}_{t}\textrm{ for all }t\in\mathbb{T}\right\}.$

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

1.

Any stopping time $\tau$ is $\mathcal{F}_{\tau}$ -measurable.
2.

If $\tau(\omega)=t\in\mathbb{T}\cup\{\infty\}$ for all $\omega\in\Omega$ then $\mathcal{F}_{\tau}=\mathcal{F}_{t}$ .
3.

If $\sigma,\tau$ are stopping times and $A\in\mathcal{F}_{\sigma}$ then $A\cap\{\sigma\leq\tau\}\in\mathcal{F}_{\tau}$ . In particular, if $\sigma\leq\tau$ then $\mathcal{F}_{\sigma}\subseteq\mathcal{F}_{\tau}$ .
4.

If $\sigma,\tau$ are stopping times and $A\in\mathcal{F}_{\sigma}$ then $A\cap\{\sigma=\tau\}\in\mathcal{F}_{\tau}$ .
5.

if the filtration $(\mathcal{F}_{t})$ is right-continuous and $\tau_{n}\geq\tau$ are stopping times with $\tau_{n}\rightarrow\tau$ then $\mathcal{F}_{\tau}=\bigcap_{n}\mathcal{F}_{\tau_{n}}$ . More generally, if $\tau_{n}=\tau$ eventually then this is true irrespective of whether the filtration is right-continuous.

If $\tau_{n}$ are stopping times with $\tau_{n}=\tau$ eventually then $\mathcal{F}_{\tau_{n}}\rightarrow\mathcal{F}_{\tau}$ . That is,

$\mathcal{F}_{\tau}=\bigcap_{n}\sigma\left(\bigcup_{m\geq n}\mathcal{F}_{\tau_{% m}}\right).$

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

$\begin{split}\displaystyle\mathcal{F}_{\tau+}&\displaystyle=\left\{A\in% \mathcal{F}_{\infty}:A\cap\{\tau\leq t\}\in\mathcal{F}_{t+}\textrm{ for every % }t\in\mathbb{T}\right\}\\ &\displaystyle=\left\{A\in\mathcal{F}_{\infty}:A\cap\{\tau<t\}\in\mathcal{F}_{% t}\textrm{ for every }t\in\mathbb{T}\right\}.\end{split}$

If $\tau_{n}\geq\tau$ are stopping times with $\tau_{n}>\tau$ whenever $\tau<\infty$ is not a maximal element of $\mathbb{T}$ , and $\tau_{n}\rightarrow\tau$ then,

$\mathcal{F}_{\tau+}=\bigcap_{n}\mathcal{F}_{\tau_{n}}=\bigcap_{n}\mathcal{F}_{% \tau_{n}+}.$

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

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

Suppose that the index set $\mathbb{T}\subseteq\mathbb{R}$ has minimal element $t_{0}$ . As the predictable $\sigma$ -algebra is generated by sets of the form $(s,\infty)\times A$ for $s\in\mathbb{T}$ and $A\in\mathcal{F}_{s}$ , and $\{t_{0}\}\times A$ for $A\in\mathcal{F}_{t_{0}}$ , the definition above can be rewritten as,

$\mathcal{F}_{\tau-}=\sigma\left(\left\{A\cap\{\tau>s\}:s\in\mathbb{T},A\in% \mathcal{F}_{s}\right\}\cup\mathcal{F}_{t_{0}}\right).$

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

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

$\mathcal{F}_{\tau-}=\sigma\left(\bigcup_{n}\mathcal{F}_{\tau_{n}}\right).$

Title	$\sigma$ -algebra at a stopping time
Canonical name	sigmaalgebraAtAStoppingTime
Date of creation	2013-03-22 18:38:56
Last modified on	2013-03-22 18:38:56
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	6
Author	gel (22282)
Entry type	Definition
Classification	msc 60G40
Related topic	DoobsOptionalSamplingTheorem

σ<math id="m1" class="ltx_Math" alttext="\sigma" display="inline"><mi>σ</mi></math>-algebra at a stopping time

$\sigma$ -algebra at a stopping time