In this post I give a proof of the theorems of optional and predictable section. These are often considered among the more advanced results in stochastic calculus, and many texts on the subject skip their proofs entirely. The approach here makes use of the measurable section theorem but, other than that, is relatively self-contained and will not require any knowledge of advanced topics beyond basic properties of probability measures.
Given a probability space we denote the projection map from to by
For a set then, by construction, for every there exists a with . Measurable section states that this choice can be made in a measurable way. That is, assuming that the probability space is complete, is measurable and there is a measurable section satisfying . I use the shorthand to mean , and it is convenient to extend the domain of to all of by setting outside of . So, we consider random times taking values in the extended nonnegative real numbers . The property that whenever can be expressed by stating that the graph of is contained in S, where the graph is defined as
The optional section theorem is a significant extension of measurable section which is very important to the general theory of stochastic processes. It starts with the concept of stopping times and with the optional sigma-algebra on . Then, it says that if S is optional its section can be chosen to be a stopping time. However, there is a slight restriction. It might not be possible to define such everywhere on , but instead only up to a set of positive probability , where can be made as small as we like. There is also a corresponding predictable section theorem, which says that if S is in the predictable sigma-algebra, its section can be chosen to be a predictable stopping time.
I give precise statements and proofs of optional and predictable section further below, and also prove a much more general section theorem which applies to any collection of random times satisfying a small number of required properties. Optional and predictable section will follow as consequences of this generalised section theorem.
Both the optional and predictable sigma-algebras, as well as the sigma-algebra used in the generalised section theorem, can be generated by collections of stochastic intervals. Any pair of random times defines a stochastic interval,
The debut of a set is defined to be the random time
In general, even if S is measurable, its debut need not be, although it can be shown to be measurable in the case that the probability space is complete. For a random time and a measurable set , we use to denote the restriction of to A defined by
We start with the general situation of a collection of random times satisfying a few required properties and show that, for sufficiently simple subsets of , the section can be chosen to be almost surely equal to the debut. It is straightforward that the collection of all stopping times defined with respect to some filtration do indeed satisfy the required properties for , but I also give a proof of this further below. A nonempty collection of subsets of a set X is called an algebra, Boolean algebra or, alternatively, a ring, if it is closed under finite unions, finite intersections, and under taking the complement of sets . Recall, also, that represents the countable intersections of A, which is the collection of sets of the form for sequences in .
Lemma 1 Let be a probability space and be a collection of measurable times satisfying,
- the constant function is in .
- and are in , for all .
- for all sequences in .
Then, letting be the collection of finite unions of stochastic intervals over , we have the following,
- is an algebra on .
- for all , its debut satisfies
and there is a with and almost surely.
Proof: By construction, the collection is closed under finite unions. Furthermore, the intersection of two stochastic intervals in
is itself a stochastic interval, so is closed under finite intersections. Noting that for any in , is the constant function equal to infinity which, therefore, is in , we see that the complement of a stochastic interval
is in . So is closed under set complements, and is an algebra.
The property is immediate from the definition of the debut and holds for any , using the fact that the infimum of a subset of is finite if and only if the set is nonempty. Similarly, for , the fact that does not require any of the properties of . The slices,
are, by construction, finite unions of left-closed intervals. Hence, is left-closed and, as any nonempty left-closed set contains its infimum, we have that is contained in S.
We now show that the debut of a set is almost surely equal to a time in . Let be the set of all with , and be the essential supremum of . By standard properties of the essential supremum, we can write for a sequence in . It follows that and , so is in . We will show that almost surely.
Write for a sequence . Choosing any n, the time
satisfies . As is in and can be expressed as a finite union ,
is in . Using , it is immediate that . Hence, is in . So, by the definition of the essential supremum, almost surely, in which case . Furthermore, whenever we necessarily have for all n and, in this case, , so .
We have constructed in such that almost surely, and whenever . It follows that the time is almost surely equal to and . To complete the proof, all that remains is to show that is in . However, this follows from writing . ⬜
In order to apply lemma 1 to more general measurable sets S, it will be necessary to approximate by sets in . This is done using the following standard lemma.
Lemma 2 Let be a finite measure space and be an algebra generating as a sigma-algebra. Then, for any and , there exists a in satisfying
Start with a sequence increasing to the limit A. By monotone convergence, we can choose n large enough that . As , there exists in such that . Then, (1) holds and, hence, A is in .
Now consider a sequence decreasing to the limit A. For each n, choose in such that . Then, setting ,
So, (1) is satisfied, and . We have shown that is closed under taking limits of increasing and decreasing sequences, and the monotone class theorem gives . ⬜
We put together the previous two lemmas to state and prove the generalised section theorem. This is the point where measurable section will be required, in order to be able to apply the previous lemma. A minor technicality is that, if we do not assume completeness of the probability space, the projection need not be measurable and so need not be in the domain of the probability measure . To get around this, we use the outer measure
which is defined on all subsets of .
Theorem 3 (Generalised Section Theorem) Let be a collection of random times defined on a probability space and satisfying the conditions of lemma 1, and let be the sigma-algebra on ,
Then, for any and , there exists a satisfying and,
Proof: Let be the collection of finite unions of stochastic intervals over . By lemma 1 this is an algebra on . As
the sigma-algebra is also generated by the stochastic intervals over , so is equal to .
The idea is to use lemma 2 in order to approximate from below by some , which requires defining a measure on . This is where we make use of the measurable section theorem, which states that there exists a random time satisfying and . A measure can then be defined by
for . This satisfies and lemma 2 states that there is an in with
Lemma 1 now gives a with
and almost surely. So,
as required. ⬜
Recall that a filtration on a probability space is a collection of sub-sigma-algebras which is increasing in t, so for . Taken together, this defines a filtered probability space . It is common to assume the usual conditions that the probability space is complete, contains all zero probability sets and that the filtration is right-continuous. We do not do this here, and will not assume any conditions other than the existence of the filtered probability space.
A stopping time is a random time such that
for all times . Defining the optional sigma-algebra on ,
the optional section theorem is as follows.
Theorem 4 (Optional Section) For any and , there exists a stopping time with and
Starting with and ,
where S is any countable dense subset of with . These sets are in , showing that and are stopping times. Similarly, if is a sequence of stopping times then,
is in , so is a stopping time. ⬜
We continue to work with respect to the filtered probability space . A map is called a predictable stopping time if there exists a sequence of stopping times increasing to and satisfying whenever . The sequence is said to announce and, as , predictable stopping times are always stopping times. The conditions are required to hold pointwise on , so that announces everywhere on . For brevity, I will also use predictable time to refer to predictable stopping times.
The predictable sigma-algebra on is defined by
Ideally, we would like to proceed in just the same way as we did above for the optional section theorem, and show that the collection of predictable times satisfies all of the required properties to apply generalised section, theorem 3. Although they very nearly satisfy the requirements, unfortunately it does not quite work. The time need not be predictable even when and are predictable, although it is almost surely equal to a predictable stopping time.
Lemma 5 The collection of predictable stopping times satisfies,
- is a predictable stopping time, for all sequences of predictable stopping times.
- for predictable stopping times and ,
- is a predictable stopping time.
- there exists a predictable stopping time with almost surely.
- for any there exists a predictable stopping time with .
Proof: Before proceeding with the proof, recall that announces if whenever . In what follows, it is convenient to relax this condition slightly so that whenever . It is still be the case that announces in the sense defined above and that is predictable.
If each () is announced by the stopping times , then it follows that the stopping times
announce which, therefore, is a predictable stopping time.
Now, let be announced, respectively, by the sequences and of stopping times. Clearly, announces which, hence, is a predictable stopping time. Next, fix an and consider the stopping times
These announce whenever is positive and , and are eventually infinite otherwise. So, they announce which is, therefore, a stopping time. By construction, and,
By monotone convergence, this tends to zero as m goes to infinity, so can be made less that any given , proving the final statement of the lemma.
Finally, we show that is almost surely equal to a predictable time. This will make use of the Borel-Cantelli lemma to construct a sequence of stopping times which almost surely announces . Start by choosing a sequence and define the times,
These are stopping times, as we see by writing
If we let A be the event that infinitely often, then it can be seen that announces which, therefore, is a predictable stopping time. Whenever , we have for large k, so and hence, .
It only remains to show that almost surely. For each k, can be chosen large enough so that
On the event we have so, from Borel-Cantelli, for large k, almost surely. This means that, up to a zero probability event, and, hence, as required. ⬜
There are several ways around the issue that need not be predictable. Lemma 5 shows that it is almost surely predictable. If we were to modify the definition of predictable stopping times so that the sequence is only required to announce almost surely, then the problem goes away. However, the definition of stopping times, predictable stopping times, and of the optional and predictable sigma algebras do not make reference to the probability measure at all, and we would prefer to keep it this way. Another approach, which amounts to the same thing, is to assume further properties of the filtration, specifically that contains all zero probability sets. We would prefer not to add further preconditions to the section theorem if they are not necessary. Another approach, taken by Dellacherie & Meyer (Probabilities and Potential, A, 1979) is to first define the predictable sigma-algebra, and then say that a stopping time is predictable if its graph is in . This also works, and such times can be shown to be announced in an almost sure sense, but we would prefer to work with the stronger and more intuitive definition given above. In fact, the predictable section theorem does hold without any such modification to the definition of predictable stopping times, and without requiring any further preconditions.
Theorem 6 (Predictable Section) For any and , there exists a predictable stopping time with and
Proof: As noted above, we cannot simply let be the collection of predictable times and apply theorem 3. The problem is that, if are in , then it does not necessarily follow that is in .
Instead, we let be the collection of stopping times such that, for all , there exists predictable stopping times and satisfying and . We show that this does satisfy the requirements stated in lemma 1.
Considering a sequence and choosing , there exists predictable times satisfying . Then,
and lemma 5 says that are predictable. Writing,
shows that as required.
Now consider , choose , so that there exists predictable times and with
Lemma 5 says that are predictable where,
shows that . Next, lemma 5 says that there are predictable stopping times satisfying
and with almost surely, and . Then,
showing that .
We have shown that satisfies all the required properties for generalised section, theorem 3, to apply. As every predictable stopping time is in , this means that for any predictable set and , there exists a time with and
Also, by definition of , there is a predictable time with . If are stopping times announcing , then announces ,
as required. ⬜