Almost Sure

29 November 16

The Section Theorems

Consider a probability space {(\Omega,\mathcal{F},{\mathbb P})} and a subset S of {{\mathbb R}_+\times\Omega}. The projection {\pi_\Omega(S)} is the set of {\omega\in\Omega} such that there exists a {t\in{\mathbb R}_+} with {(t,\omega)\in S}. We can ask whether there exists a map

\displaystyle  \tau\colon\pi_\Omega(S)\rightarrow{\mathbb R}_+

such that {(\tau(\omega),\omega)\in S}. From the definition of the projection, values of {\tau(\omega)} satisfying this exist for each individual {\omega}. By invoking the axiom of choice, then, we see that functions {\tau} with the required property do exist. However, to be of use for probability theory, it is important that {\tau} should be measurable. Whether or not there are measurable functions with the required properties is a much more difficult problem, and is answered affirmatively by the measurable selection theorem. For the question to have any hope of having a positive answer, we require S to be measurable, so that it lies in the product sigma-algebra {\mathcal{B}({\mathbb R}_+)\otimes\mathcal{F}}, with {\mathcal{B}({\mathbb R}_+)} denoting the Borel sigma-algebra on {{\mathbb R}_+}. Also, less obviously, the underlying probability space should be complete. Throughout this post, {(\Omega,\mathcal{F},{\mathbb P})} will be assumed to be a complete probability space.

It is convenient to extend {\tau} to the whole of {\Omega} by setting {\tau(\omega)=\infty} for {\omega} outside of {\pi_\Omega(S)}. Then, {\tau} is a map to the extended nonnegative reals {\bar{\mathbb R}_+={\mathbb R}_+\cup\{\infty\}} for which {\tau(\omega) < \infty} precisely when {\omega} is in {\pi_\Omega(S)}. Next, the graph of {\tau}, denoted by {[\tau]}, is defined to be the set of {(t,\omega)\in{\mathbb R}_+\times\Omega} with {t=\tau(\omega)}. The property that {(\tau(\omega),\omega)\in S} whenever {\tau(\omega) < \infty} is expressed succinctly by the inclusion {[\tau]\subseteq S}. With this notation, the measurable selection theorem is as follows.

Theorem 1 (Measurable Selection) For any {S\in\mathcal{B}({\mathbb R}_+)\otimes\mathcal{F}}, there exists a measurable {\tau\colon\Omega\rightarrow\bar{\mathbb R}_+} such that {[\tau]\subseteq S} and

\displaystyle  \left\{\tau < \infty\right\}=\pi_\Omega(S). (1)

As noted above, if it wasn’t for the measurability requirement then this theorem would just be a simple application of the axiom of choice. Requiring {\tau} to be measurable, on the other hand, makes the theorem much more difficult to prove. For instance, it would not hold if the underlying probability space was not required to be complete. Note also that, stated as above, measurable selection implies that the projection of S is equal to a measurable set {\{\tau < \infty\}}, so the measurable projection theorem is an immediate corollary. I will leave the proof of Theorem 1 for a later post, together with the proofs of the section theorems stated below.

A closely related problem is the following. Given a measurable space {(X,\mathcal{E})} and a measurable function, {f\colon X\rightarrow\Omega}, does there exist a measurable right-inverse on the image of {f}? This is asking for a measurable function, {g}, from {f(X)} to {X} such that {f(g(\omega))=\omega}. In the case where {(X,\mathcal{E})} is the Borel space {({\mathbb R}_+,\mathcal{B}({\mathbb R}_+))}, Theorem 1 says that it does exist. If S is the graph {\{(t,f(t))\colon t\in{\mathbb R}_+\}} then {\tau} will be the required right-inverse. In fact, as all uncountable Polish spaces are Borel-isomorphic to each other and, hence, to {{\mathbb R}_+}, this result applies whenever {(X,\mathcal{E})} is a Polish space together with its Borel sigma-algebra.

Continuous-time stochastic calculus is generally applied under the setting of a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in{\mathbb R}_+},{\mathbb P})} which, as usual, I am assuming to be complete. It can be important to choose the random time {\tau} given by Theorem 1 to be a stopping time. For this to be possible, the stronger measurability criterion that S is in the optional sigma-algebra {\mathcal{O}} is required.

Theorem 2 (Optional Section) For any {S\in\mathcal{O}} and {\epsilon > 0}, there exists a stopping time {\tau} such that {[\tau]\subseteq S} and

\displaystyle  {\mathbb P}\left(\tau < \infty\right) > {\mathbb P}\left(\pi_\Omega(S)\right)-\epsilon. (2)

In the statement of the optional section theorem, the equality (1) has been replaced by an inequality (2). The condition {[\tau]\subseteq S} ensures that {\{\tau < \infty\}} is a subset of {\pi_\Omega(S)}, and the measure of the difference of these events is

\displaystyle  {\mathbb P}\left(\pi_\Omega(S)\setminus\{\tau < \infty\}\right) ={\mathbb P}\left(\pi_\Omega(S)\right)-{\mathbb P}\left(\tau < \infty\right).

Inequality (2) states that this can be made as small as we like and, so, is an approximate form of equality (1).

In fact, it is not generally possible to choose the stopping time {\tau} to satisfy (1). To see this, consider the following example. Let {\sigma} be a random variable taking values in {(0, \infty)} and such that {{\mathbb P}(\sigma < t) > 0} for each positive t. For example, take {\sigma} to be uniformly distributed on {(0,1)}. Let {\{\mathcal{F}_t\}_{t\in{\mathbb R}_+}} be the (completed) natural filtration of {1_{[\sigma,\infty)}}, so that {\sigma} is a stopping time, and set {S=(0,\sigma)}. It can be seen that {\mathcal{F}_t} is generated by the sets {\{\sigma\le s\}} over {s\le t} and, hence, is trivial when restricted to {\sigma > t}. So, every {\mathcal{F}_t}-measurable random variable is almost-surely constant on the event {\{\sigma > t\}}. Now, any stopping time {\tau} can be seen to be deterministic on the event {\tau < \sigma}. So, if {[\tau]\subseteq S}, we have

\displaystyle  \tau = \begin{cases} s,&\textrm{if }\sigma > s,\\ \infty,&\textrm{if }\sigma\le s. \end{cases}

almost surely, for some fixed positive s. Then,

\displaystyle  {\mathbb P}(\tau < \infty)={\mathbb P}(\sigma > s) < 1={\mathbb P}(\pi_\Omega(S)).

Next, for a predictable set, {S\in\mathcal{P}}, it is possible to do better than the optional section theorem. The stopping time {\tau} can be chosen to be predictable.

Theorem 3 (Predictable Section) For any {S\in\mathcal{P}} and {\epsilon > 0}, there exists a predictable stopping time {\tau} such that {[\tau]\subseteq S} and

\displaystyle  {\mathbb P}\left(\tau < \infty\right) > {\mathbb P}\left(\pi_\Omega(S)\right)-\epsilon.

Again, it is not generally possible to choose {\tau} to satisfy (1). With the stopping time {\sigma} and filtration {\{\mathcal{F}_t\}_{t\in{\mathbb R}_+}} as in the example above, consider taking {S=(0,\sigma]\in\mathcal{P}}. Let {\tau} be a predictable stopping time with {[\tau]\subseteq S}. If {\{\tau_n\}_{n=1,2,\ldots}} is a sequence of stopping times announcing {\tau} then, without loss of generality, we can set {\tau_n=\infty} whenever {\tau_n\ge\sigma}. Using the argument above, there are deterministic times {s_n\ge0} such that {\tau_n=s_n} (a.s.) whenever {\sigma > s_n}. Taking the limit as n goes to infinity, there exists a deterministic time s with

\displaystyle  \tau = \begin{cases} s,&\textrm{if }\sigma \ge s,\\ \infty,&\textrm{if }\sigma < s. \end{cases}

almost surely. Then, again,

\displaystyle  {\mathbb P}(\tau < \infty)={\mathbb P}(\sigma \ge s) < 1={\mathbb P}(\pi_\Omega(S)).

A straightforward consequence of the measurable selection and section theorems is that, to test whether two processes are equal, it is enough to compare them at each random time, up to zero probability sets. Throughout, I am considering processes to be equal if they are equal up to evanescence.

Theorem 4 Let X and Y be stochastic processes.

  • Measurable Selection: If X,Y are jointly measurable and {X_\tau=Y_\tau} (a.s.) for each {\mathcal{F}}-measurable random time {\tau}, then {X=Y}.
  • Optional Section: If X,Y are optional and {X_\tau=Y_\tau} (a.s.) for each stopping time {\tau}, then {X=Y}.
  • Predictable Section: If X,Y are predictable and {X_\tau=Y_\tau} (a.s.) for each predictable stopping time {\tau}, then {X=Y}.

Proof: Let {S=\{X\not=Y\}}. That is, S is the set of {(t,\omega)} for which {X_t(\omega)\not=Y_t(\omega)}, and lies in the sigma-algebra generated by X and Y. Any random time {\tau} with graph contained in S satisfies {X_\tau\not=Y_\tau} whenever {\tau < \infty}. I will argue by contradiction, so suppose that {X\not=Y}. Then, S is not evanescent, hence {{\mathbb P}(\pi_\Omega(S)) > 0}.

If X,Y are jointly measurable then so is S and, by Theorem 1, there exists a random time {\tau} with graph contained in S and,

\displaystyle  {\mathbb P}(X_\tau\not=Y_\tau)={\mathbb P}(\tau < \infty) > 0,

contradicting the condition of the first statement.

Next, if X,Y are optional then so is S and, by Theorem 2, there exists a stopping time {\tau} with graph contained in S and,

\displaystyle  {\mathbb P}(X_\tau\not=Y_\tau)={\mathbb P}(\tau < \infty) > {\mathbb P}(\pi_\Omega(S))-\epsilon. (3)

Taking {\epsilon} small enough that the right hand side is non-negative contradicts the condition of the second statement.

Finally, if X,Y are predictable then so is S. Theorem 3 states that there exists a predictable stopping time {\tau} with graph contained in S, and such that (3) holds. This contradicts the condition of the third statement. ⬜

We previously showed that a map {\tau\colon\Omega\rightarrow\bar{\mathbb R}_+} is measurable if and only if {[\tau]} is jointly measurable, and is a stopping time if {[\tau]} is progressive or, equivalently, is optional. In the previous post, it was also mentioned that {\tau} is a predictable stopping time if and only if {[\tau]} is a predictable set. However, advanced techniques are required to prove this statement, so no proof was given at the time. With the aid of the predictable section theorem, I will now give a proof.

Lemma 5 A map {\tau\colon\Omega\rightarrow\bar{\mathbb R}_+} is a predictable stopping time if and only if {[\tau]\in\mathcal{P}}.

Proof: By predictable section, Theorem 3, there are predictable stopping times {\{\tau_n\}_{n=1,2,\ldots}} such that {[\tau_n]\subseteq[\tau]} and

\displaystyle  {\mathbb P}(\tau_n\not=\tau) = {\mathbb P}(\tau < \infty)-{\mathbb P}(\tau_n < \infty) < 2^{-n}.

By the Borel-Cantelli lemma, {\tau_n=\tau} for all large enough n, almost surely. Hence, {\tau} is a predictable stopping time. ⬜

Compare this to the classification of predictable stopping times previously proven in these notes. There, a stopping time was shown to be predictable if and only if its graph is predictable. The proof used there relied on some advanced properties of the stochastic integral in order to show that {[\tau]\in\mathcal{P}} implies that {\tau} is fair. This required constructing the paths of stochastic integrals of {1_{[\tau]}} with respect to bounded martingales. Although it requires predictable section, the proof given in this post is more direct, and is also much stronger, since it does not require {\tau} to already be known to be a stopping time.

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2 Comments »

  1. Not exactly a comment but an answer to your (George Lowther) request on RG, who don’t accept djvu (why?). You will find many of my books
    (in djvu) on the russian library “genesis”. Here is one address (many mirrors): http://gen.lib.rus.ec/
    Best regards
    CD

    Comment by Anonymous — 30 November 16 @ 11:20 AM | Reply

    • Many thanks for that, I was able to find the book I was looking for (although my IP has actually blocked that website).

      Incidentally, it’s nice to get a personal response from such a master and authority on the subject!

      Regards
      George Lowther

      Comment by George Lowther — 30 November 16 @ 5:01 PM | Reply


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