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.

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\{\tau<t\}} 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. ⬜

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. ⬜

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<t<\infty{\rm\ or\ }\sigma=0\}}+1_{A\cap\{t=\infty\}}

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. ⬜

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<t\})\in\mathcal{F}_t.

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<t+1/n\})\in\mathcal{F}_{t+}=\mathcal{F}_t

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. ⬜

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<t, \end{cases}

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. ⬜

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. For a sigma-algebra {\mathcal E} on {\Omega}, and a subset {S\subseteq \Omega}, we use {\mathcal E\vert_S} to denote the sigma-algebra on S consisting of sets {S\cap A} for {A\in\mathcal E}.

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

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\mathcal{F}_\sigma\vert_{\{\sigma=\tau\}}=\mathcal{F}_{\tau}\vert_{\{\sigma=\tau\}},\smallskip\\ &\displaystyle\mathcal{F}_{\sigma-}\vert_{\{\sigma=\tau\}}=\mathcal{F}_{\tau-}\vert_{\{\sigma=\tau\}}. \end{array}

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}.

Next, {\mathcal F_{\sigma-}} is generated by sets in {\mathcal F_0\subseteq \mathcal F_{\tau-}}, and by sets of the form {A\cap\{t < \sigma\}} for {A\in\mathcal F_t} which, restricted to {\{\sigma=\tau\}}, coincide with {A\cap\{t < \tau\}\in\mathcal F_{\tau-}}. So, {\mathcal F_{\sigma-}\vert_{\{\sigma=\tau\}}\subseteq\mathcal F_{\tau-}\vert_{\{\sigma=\tau\}}}, and the reverse inclusion follows by exchanging {\sigma} and {\tau}. ⬜

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. ⬜

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}_+}. ⬜

9 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 | Reply

  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 | Reply

    • You’re right. I fixed it, thanks.

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

      • 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 | Reply

  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 | Reply

  4. Hi, I am struggling with the definition of the “stopping-time sigma-algebra” (aka sigma algebra at stopping time). You state that, “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.” Is it possible to get a concrete example please? Thanks.

    Comment by HW — 17 January 18 @ 5:53 PM | Reply

  5. Hi, Thanks for the excellent posts. One question: Why is the sigma algebra at a stopping time “tau” not defined as the sigma algebra generated by tau, in the first place? How is this notion related to the (standard) definition that you have given above? I vaguely recall some mention (not sure in which reference) that the former definition is not operationally useful to perform the related analysis–is that so or are there other reasons?

    Comment by RB — 24 November 18 @ 12:02 PM | Reply

    • Consider the case where \tau is constant, equal to a deterministic time t\in\mathbb{R}_+. Then \mathcal{F}_\tau=\mathcal{F}_t. With the definition you suggest, \mathcal{F}_\tau would be trivial. More generally, the sigma algebra generated by \tau would miss out lots of events observable at the stopping time. We want X_\tau to be \mathcal{F}_\tau-measurable for any reasonably regular adapted process X.

      Comment by George Lowther — 6 December 18 @ 10:56 PM | Reply


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