# Almost Sure

## 23 November 09

### Sigma Algebras at a Stopping Time

The previous post introduced the notion of a stopping time ${\tau}$. A stochastic process ${X}$ can be sampled at such random times and, if the process is jointly measurable, ${X_\tau}$ will be a measurable random variable. It is usual to study adapted processes, where ${X_t}$ is measurable with respect to the sigma-algebra ${\mathcal{F}_t}$ at that time. Then, it is natural to extend the notion of adapted processes to random times and ask the following. What is the sigma-algebra of observable events at the random time ${\tau}$, and is ${X_\tau}$ measurable with respect to this? The idea is that if a set ${A}$ is observable at time ${\tau}$ then for any time ${t}$, its restriction to the set ${\{\tau\le t\}}$ should be in ${\mathcal{F}_t}$. As always, we work with respect to a filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$. The sigma-algebra at the stopping time ${\tau}$ is then,

$\displaystyle \mathcal{F}_\tau=\left\{A\in\mathcal{F}_\infty\colon A\cap\{\tau\le t\}\in\mathcal{F}_t{\rm\ for\ all\ }t\ge 0\right\}.$

The restriction to sets in ${\mathcal{F}_\infty}$ is to take account of the possibility that the stopping time can be infinite, and it ensures that ${A=A\cap\{\tau\le\infty\}\in\mathcal{F}_\infty}$. From this definition, a random variable ${U}$ us ${\mathcal{F}_\tau}$-measurable if and only if ${1_{\{\tau\le t\}}U}$ is ${\mathcal{F}_t}$-measurable for all times ${t\in{\mathbb R}_+\cup\{\infty\}}$.

Similarly, we can ask what is the set of events observable strictly before the stopping time. For any time ${t}$, then this sigma-algebra should include ${\mathcal{F}_t}$ restricted to the event ${\{t<\tau\}}$. This suggests the following definition,

$\displaystyle \mathcal{F}_{\tau-}=\sigma\left(\left\{ A\cap\{t<\tau\}\colon t\ge 0,A\in\mathcal{F}_t \right\}\cup\mathcal{F}_0\right).$

The notation ${\sigma(\cdot)}$ denotes the sigma-algebra generated by a collection of sets, and in this definition the collection of elements of ${\mathcal{F}_0}$ are included in the sigma-algebra so that we are consistent with the convention ${\mathcal{F}_{0-}=\mathcal{F}_0}$ used in these notes.

With these definitions, the question of whether or not a process ${X}$ is ${\mathcal{F}_\tau}$-measurable at a stopping time ${\tau}$ can be answered. There is one minor issue here though; stopping times can be infinite whereas stochastic processes in these notes are defined on the time index set ${{\mathbb R}_+}$. We could just restrict to the set ${\{\tau<\infty\}}$, but it is handy to allow the processes to take values at infinity. So, for the moment we consider a processes ${X_t}$ where the time index ${t}$ runs over ${\bar{\mathbb R}_+\equiv{\mathbb R}_+\cup\{\infty\}}$, and say that ${X}$ is a predictable, optional or progressive process if it satisfies the respective property restricted to times in ${{\mathbb R}_+}$ and ${X_\infty}$ is ${\mathcal{F}_\infty}$-measurable.

Lemma 1 Let ${X}$ be a stochastic process and ${\tau}$ be a stopping time.

• If ${X}$ is progressively measurable then ${X_\tau}$ is ${\mathcal{F}_\tau}$-measurable.
• If ${X}$ is predictable then ${X_\tau}$ is ${\mathcal{F}_{\tau-}}$-measurable.

Proof: If ${X}$ is progressive then, as proven in the previous post, the stopped process ${X^\tau}$ is also progressive and, hence, is adapted. It follows that ${1_{\{\tau\le t\}}X_\tau=1_{\{\tau\le t\}}X^\tau_t}$ is ${\mathcal{F}_t}$-measurable which, from the definition above, implies that ${1_{\{\tau<\infty\}}X_\tau}$ is ${\mathcal{F}_{\tau}}$-measurable.

Furthermore, ${1_{\{\tau=\infty\}}X_\tau}$ is ${\mathcal{F}_\infty}$-measurable and is zero when restricted to the set ${{\{\tau\le t\}}}$ for all ${t\in{\mathbb R}_+}$, so is also ${\mathcal{F}_\tau}$-measurable.

Now, consider a predictable process ${X}$. Write ${\mathcal{\bar P}}$ for the predictable sigma-algebra on ${\bar{\mathbb R}_+\times\Omega}$. That is, the subsets of ${S\subseteq\bar{\mathbb R}_+\times\Omega}$ which are predictable when restricted to ${{\mathbb R}_+\times\Omega}$ and such that ${\{\omega\in\Omega\colon(\infty,\omega)\in S\}}$ is ${\mathcal{F}_\infty}$-measurable. Then, ${X}$ is ${\mathcal{\bar P}}$-measurable. By the functional monotone class theorem, it is enough to prove the result for processes of the form ${X_t(\omega)=1_{\{(t,\omega)\in S\}}}$ for some pi-system of sets generating ${\mathcal{\bar P}}$.

The predictable sigma algebra is generated by the sets ${S}$ of the following forms,

1. ${S=(t,\infty]\times A}$ for times ${t\in{\mathbb R}_+}$ and ${A\in\mathcal{F}_t}$. If ${X=1_S}$ then ${ X_\tau=1_{\{t<\tau\}\cap A} }$ which, by definition, is ${\mathcal{F}_{\tau-}}$-measurable.
2. ${S=\{0\}\times A}$ for ${A\in\mathcal{F}_0}$. If ${X=1_S}$ then ${ X_\tau= 1_{\{\tau=0\}\cap A} }$ which is ${\mathcal{F}_0}$-measurable, and so is also ${\mathcal{F}_{\tau-}}$-measurable.

$\Box$

So, the `adaptedness’ of measurable processes extends to stopping times. In fact, it is possible to go further and use this as an alternative definition of these sigma-algebras.

Lemma 2 Let ${U}$ be a random variable and ${\tau}$ be a stopping time. Then,

• ${U}$ is ${\mathcal{F}_\tau}$-measurable if and only if ${U=X_\tau}$ for some progressively measurable (resp. optional) process ${X}$.
• ${U}$ is ${\mathcal{F}_{\tau-}}$-measurable if and only if ${U=X_{\tau}}$ for some predictable process ${X}$.

Proof: If ${U}$ is ${\mathcal{F}_\tau}$-measurable, then the process ${X_{t}=1_{\{t\ge\tau\}}U}$ is adapted and right-continuous. Therefore, it is optional (and hence, progressive) and clearly ${U=X_\tau}$.

For the second statement, consider the set ${V}$ of random variables which can be expressed as ${X_\tau}$ for a predictable process ${X}$. The functional monotone class theorem can be used to show that ${V}$ contains all ${\mathcal{F}_{\tau-}}$-measurable random variables. First, ${V}$ is clearly closed under taking linear combinations. Second, if ${U_n\in V}$ is increasing to the limit ${U}$ then there exists predictable processes ${X^n}$ with ${U_n=X^n_\tau}$. Then, ${U=\limsup_nX^n_\tau}$ is also in ${V}$.

Finally, it just needs to be shown that ${1_S\in V}$ for all ${S}$ in a pi-system generating ${\mathcal{F}_{\tau-}}$. By definition, the following sets generate ${\mathcal{F}_{\tau-}}$.

• ${S\in\mathcal{F}_0}$. In this case, ${1_S=X_\tau}$ with ${X_t(\omega)=1_{\{\omega\in S\}}}$.
• ${S=A\cap\{t<\tau\}}$ for ${t\in{\mathbb R}_+}$ and ${A\in\mathcal{F}_t}$. Then, ${1_S=X_\tau}$ with ${X_s(\omega)=1_{\{s>t,\omega\in A\}}}$.

In both these cases, ${X}$ is left-continuous and adapted and, hence, is predictable. $\Box$

This result gives the main motivation for the definitions of ${\mathcal{F}_\tau}$ and ${\mathcal{F}_{\tau-}}$. For the remainder of this post, I state and prove several simple results which are useful for general applications of stopping times.

Lemma 3 Any stopping time ${\tau}$ is both ${\mathcal{F}_{\tau}}$ and ${\mathcal{F}_{\tau-}}$-measurable.

Proof: The deterministic process ${X_t\equiv t}$ is trivially adapted and both left and right-continuous, so it is predictable and optional. Consequently, by the previous lemma, ${\tau=X_{\tau}}$ is ${\mathcal{F}_\tau}$ and ${\mathcal{F}_{\tau-}}$-measurable. $\Box$

Next, the sigma-algebras are increasing in the sense that we would hope.

Lemma 4 For any stopping time ${\tau}$,

$\displaystyle \mathcal{F}_{\tau-}\subseteq\mathcal{F}_\tau.$

If ${\sigma\le\tau}$ is any other stopping time then,

$\displaystyle \mathcal{F}_\sigma\subseteq\mathcal{F}_\tau,\ \mathcal{F}_{\sigma-}\subseteq\mathcal{F}_{\tau-}.$

If, furthermore, ${\sigma<\tau}$ whenever ${\tau\not\in\{0,\infty\}}$ then ${\mathcal{F}_\sigma\subseteq\mathcal{F}_{\tau-}}$.

Proof: This proof makes use of Lemma 2. First, by the lemma, every ${\mathcal{F}_{\tau-}}$-measurable set ${A}$ can be written in the form ${1_A=X_\tau}$ for a predictable process ${X}$. However, as predictable processes are progressive, ${A}$ will also be in ${\mathcal{F}_\tau}$.

Now suppose that ${A}$ is ${\mathcal{F}_\sigma}$ (resp. ${\mathcal{F}_{\sigma-}}$) measurable. Then, there is a progressive (resp. predictable) process satisfying ${X_\tau=1_A}$. As the stopped process ${X^\sigma}$ is also progressive (resp. predictable) it follows that ${1_A=X^\sigma_\tau}$ is ${\mathcal{F}_\tau}$ (resp. ${\mathcal{F}_{\tau-}}$)-measurable.

Finally, suppose that ${\sigma<\tau}$ whenever ${\tau\not\in\{0,\infty\}}$ and that ${A\in\mathcal{F}_\sigma}$. Then

$\displaystyle X_t\equiv 1_{A\cap\{\sigma

is a left-continuous and adapted at finite times, and ${X_\infty}$ is ${\mathcal{F}_\infty}$-measurable. Hence, it is predictable process and ${1_A=X_\tau}$ is ${\mathcal{F}_{\tau-}}$-measurable. $\Box$

The sigma-algebras satisfy the expected left and right-limits. In the following lemma, the first statement says that right-continuity of a filtration extends to arbitrary stopping times. The second says that ${\mathcal{F}_{\tau-}}$ can indeed be interpreted as a left-limit. However, this statement does not say anything much for arbitrary stopping times, because it is not in general possible to strictly approximate them from the left in this way. If such a sequence ${\tau_n}$ does indeed exist then the stopping time is called predictable.

Lemma 5 Let ${\tau_n\rightarrow\tau}$ be stopping times. Then

• If the filtration ${\{\mathcal{F}_t\}}$ is right-continuous and ${\tau_n\ge\tau}$ for each ${n}$ then

$\displaystyle \bigcap_n\mathcal{F}_{\tau_n}=\mathcal{F}_\tau.$

• If ${\tau_n\le\tau}$, with a strict inequality whenever ${\tau\not\in\{0,\infty\}}$, then

$\displaystyle \sigma\left(\bigcup_n\mathcal{F}_{\tau_n}\right) =\sigma\left(\bigcup_n\mathcal{F}_{\tau_n-}\right)=\mathcal{F}_{\tau-}.$

Proof: Starting with the first statement, we know that ${\mathcal{F}_\tau\subseteq\mathcal{F}_{\tau_n}}$. So, it just needs to be shown that any ${A\in\cap_n\mathcal{F}_{\tau_n}}$ is in ${\mathcal{F}_\tau}$. Any such set satisfies

$\displaystyle A\cap\{\tau< t\} = \bigcup_n(A\cap\{\tau_n

Then, by right-continuity of the filtration, for any ${m\ge 1}$

$\displaystyle A\cap\{\tau\le t\}=\bigcap_{n=m}^\infty(A\cap\{\tau

as required.

For the second statement, we know that ${\mathcal{F}_{\tau_n-}\subseteq\mathcal{F}_{\tau_n}\subseteq\mathcal{F}_{\tau-}}$, so it is only necessary to prove that there is a generating set for ${\mathcal{F}_{\tau-}}$ lying in ${\mathcal{G}\equiv\sigma(\cup_n\mathcal{F}_{\tau_n-})}$. As ${\mathcal{F}_0\subseteq\mathcal{G}}$ it is enough to consider sets of the form ${S=A\cap\{t<\tau\}}$ for ${A\in\mathcal{F}_t}$. However

$\displaystyle A\cap\{t<\tau\}=\bigcup_n(A\cap\{t<\tau_n\})\in\mathcal{G}$

as required. $\Box$

As should be the case, the definition of the sigma-algebras at a constant stopping time is consistent with the filtration.

Lemma 6 If ${\tau\colon\Omega\rightarrow\bar{\mathbb R}_+}$ is equal to the constant value ${t}$ then,

$\displaystyle \mathcal{F}_\tau=\mathcal{F}_t,\ \mathcal{F}_{\tau-}=\mathcal{F}_{t-}.$

Proof: If ${A\in\mathcal{F}_t}$ then for all times ${s}$,

$\displaystyle A\cap\{\tau\le s\}=\begin{cases} A\in\mathcal{F}_t\subseteq\mathcal{F}_s,&\textrm{if }s\ge t,\\ \emptyset\in\mathcal{F}_s,&\textrm{if }s

showing that ${A\in\mathcal{F}_\tau}$. Conversely, if ${A\in\mathcal{F}_\tau}$ then ${A=A\cap\{\tau\le t\}\in\mathcal{F}_t}$ as required.

This shows that ${\mathcal{F}_{\tau}=\mathcal{F}_t}$. The equality ${\mathcal{F}_{\tau-}=\mathcal{F}_{t-}}$ follows by taking left limits and applying the previous lemma. $\Box$

Given two stopping times ${\sigma,\tau}$ it follows from the definitions that ${\{\sigma\le\tau\}\in\mathcal{F}_\tau}$ and ${\{\sigma<\tau\}\in\mathcal{F}_{\tau-}}$. So, ${\{\sigma=\tau\}}$ is in ${\mathcal{F}_\tau}$ and, by symmetry, is also in ${\mathcal{F}_\sigma}$. Furthermore, these two sigma-algebras coincide when restricted to this set.

Lemma 7 If ${\sigma,\tau}$ are stopping times then

$\displaystyle \mathcal{F}_\sigma\vert_{\{\sigma=\tau\}}=\mathcal{F}_{\tau}\vert_{\{\sigma=\tau\}}.$

Proof: If ${A\subseteq\{\sigma=\tau\}}$ is ${\mathcal{F}_\sigma}$ measurable then ${A\cap\{\tau\le t\}=A\cap\{\sigma\le t\}}$ is in ${\mathcal{F}_t}$ and ${A\in\mathcal{F}_\tau}$. The reverse inclusion follows by exchanging ${\sigma}$ and ${\tau}$. $\Box$

Given a stopping time taking values in a countable set of times, the following result is often useful to show that a set is in the sigma algebra by checking it at each of the fixed times.

Lemma 8 Let ${\tau,(\tau_n)_{n=1,2,\ldots}}$ be stopping times such that ${\tau(\omega)\in\{\tau_1(\omega),\tau_2(\omega),\ldots\}}$ for all ${\omega\in\Omega}$.

A set ${A\subseteq\Omega}$ is in ${\mathcal{F}_\tau}$ if and only if ${A\cap\{\tau_n=\tau\}\in\mathcal{F}_{\tau_n}}$ for each ${n}$.

Proof: By the previous lemma, if ${A\in\mathcal{F}_\tau}$ then ${A\cap\{\tau=\tau_n\}\in\mathcal{F}_{\tau_n}}$. Conversely,

$\displaystyle A\cap\{\tau\le t\}=\bigcup_n((A\cap\{\tau=\tau_n\})\cap\{\tau_n\le t\})\in\mathcal{F}_t$

as required. $\Box$

Finally, the following result is used to construct new stopping times out of old ones. If we wait until a time ${\tau}$ occurs, and then decide to either use that time or not based on an ${\mathcal{F}_\tau}$-measurable event, the result is again a stopping time.

Lemma 9 Let ${\tau}$ be a stopping time and ${A\in\mathcal{F}_\tau}$. Then,

$\displaystyle \tau_A(\omega)\equiv\begin{cases} \tau(\omega),&\textrm{if }\omega\in A,\\ \infty,&\textrm{otherwise} \end{cases}$

is also a stopping time.

Proof: This follows from the following

$\displaystyle \left\{\tau_A\le t\right\}=A\cap\left\{\tau\le t\right\}\in\mathcal{F}_t$

for all ${t\in{\mathbb R}_+}$. $\Box$

## 6 Comments »

1. Dear Almost sure,
In lemma 7, do you want to show
$... = \mathcal{F}_\tau |_{\{\sigma = \tau\}}$
Thanks

Comment by kenneth — 30 April 10 @ 4:02 PM

2. Dear Georges,
Excuse me if I am wrong, but in the proof of Lemma 2, last but one line, shouldn’t it be $S=A\cap\{t <\tau \}$ ?

Comment by josh — 3 October 12 @ 9:40 AM

• You’re right. I fixed it, thanks.

Comment by George Lowther — 3 October 12 @ 11:32 AM

• Thanks for your answer. Did you also by any chance see my question on the Stochastic Integral ?

Comment by josh — 8 October 12 @ 9:40 AM

3. Hello George,
I recently meet the problem on consistency of probability measures. Given a sequence of probability measures Q_n, how is it possible to check that they are consistent? The definition seems impractical. What I want is a method that we can actually apply facing concrete examples. Also, where shall I start if I want to find counter-examples?
Thank you very much!

Comment by Zhenyu (Rocky) Cui — 1 November 12 @ 7:55 PM

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