In this post, I introduce the concept of optional and predictable projections of jointly measurable processes. Optional projections of right-continuous processes and predictable projections of left-continuous processes were constructed in earlier posts, with the respective continuity conditions used to define the projection. These are, however, just special cases of the general theory. For arbitrary measurable processes, the projections cannot be expected to satisfy any such pathwise regularity conditions. Instead, we use the measurability criteria that the projections should be, respectively, optional and predictable.
The projection theorems are a relatively straightforward consequence of optional and predictable section. However, due to the difficulty of proving the section theorems, optional and predictable projection is generally considered to be an advanced or hard part of stochastic calculus. Here, I will make use of the section theorems as stated in an earlier post, but leave the proof of those until after developing the theory of projection.
As usual, we work with respect to a complete filtered probability space , and only consider real-valued processes. Any two processes are considered to be the same if they are equal up to evanescence. The optional projection is then defined (up to evanescence) by the following.
Theorem 1 (Optional Projection) Let X be a measurable process such that is almost surely finite for each stopping time . Then, there exists a unique optional process , referred to as the optional projection of X, satisfying
almost surely, for each stopping time .
Predictable projection is defined similarly.
Theorem 2 (Predictable Projection) Let X be a measurable process such that is almost surely finite for each predictable stopping time . Then, there exists a unique predictable process , referred to as the predictable projection of X, satisfying
almost surely, for each predictable stopping time .
When the integrability criterion of Theorem 1 or Theorem 2 is satisfied then I will simply say that the optional projection or, respectively, the predictable projection exists. I will give the proofs of these theorems in a moment but, first, state some equivalent forms.
Theorem 3 The optional projection of a measurable process X exists if and only if is almost surely finite for each bounded stopping time . Then, is the unique optional process satisfying
almost surely, for each bounded stopping time .
Proof: From the definition, if the optional projection exists then is almost surely finite for every stopping time and, in particular, for every bounded stopping time. Conversely, for any fixed time T, and agree on the event . So,
is almost surely finite. Letting T increase to infinity, is almost surely finite as required.
Predictable projection can be defined similarly.
Theorem 4 The predictable projection of X exists if and only if is almost surely finite for each bounded predictable stopping time . Then, is the unique predictable process satisfying
almost surely, for each bounded predictable stopping time .
Proof: The proof is exactly the same as that given for Theorem 3, only with `stopping time’ replaced by `predictable stopping time’ and replaced by . ⬜
Next, restricting to processes whose values at stopping times are integrable, optional projection can be defined by taking expectations rather than conditional expectations.
Theorem 5 Let X be a measurable process such that is integrable for each stopping time . Then, the optional projection, , exists and is the unique optional process such that, for each stopping time , is integrable and satisfies
Similarly, for predictable projection.
Theorem 6 Let X be a measurable process such that is integrable for each predictable stopping time . Then, the predictable projection, , exists and is the unique predictable process such that, for each predictable stopping time , is integrable and satisfies
Proof: The proof is exactly the same as that given for Theorem 5, with `stopping time’ replaced by `predictable stopping time’ and replaced by . The fact that is a predictable stopping time when is also required. ⬜
Existence of optional and predictable projections
If they exist, then the uniqueness of optional and predictable projections is an immediate consequence of the section theorems. This is because (1) and (2) almost surely determine the values of the optional process and predictable process at every stopping time and, respectively, at every predictable stopping time, uniquely identifying them up to evanescence. In contrast, proving existence requires a bit more work, but only involves the basic theory of continuous martingales. I start with the projections of constant processes. Under the usual conditions, this makes use of the existence of cadlag versions of martingales. As I am not assuming right-continuity of the filtration, we need to relax the cadlag condition a bit to include martingales which are right-continuous outside of a countable set of times.
Lemma 7 Let X be the constant process , for an integrable random variable U. Let M be a version of the martingale
which has right and left limits everywhere, and is right-continuous outside of a countable set . Such versions always exist.
Then, the optional and predictable projections of X exist, and are given by
Proof: Start by considering a simple stopping time, , taking values in for a finite set . Then,
is satisfied at simple stopping times. Now, for an arbitrary stopping time , let be a sequence of simple stopping times decreasing to and with eventually whenever . For example, let be a sequence of finite subsets of increasing to a dense subset of containing S, and set
As the conditional expectations given by (7) are uniformly integrable, and , we can take limits,
So, as in the proof of Theorem 5, as required.
Next, if is a predictable stopping time then choose a sequence of stopping times announcing . So, and, for each fixed time T,
Letting T increase to infinity gives , so as required. ⬜
The completion of the proof of existence of optional and predictable projections is now just an application of the monotone class theorem.
Proof of Theorem 1: Let denote the space of measurable processes, X, such that is almost surely finite for every stopping time , and let denote the set of such that there exists an optional process, , satisfying (1). We just need to show that . The monotone class theorem implies this, so long as we can verify the following.
- There exists a pi-system generating the sigma-algebra such that for all .
- is closed under finite linear combinations.
- For any sequence of nonnegative increasing to a limit , then .
For the first statement above, let denote the products for and . By Lemma 7, the constant process has an optional projection. So, has optional projection , and is in .
For the second statement above, it is clear that the optional projection of a linear combination of processes is equal to the linear combination of their optional projections.
For the third statement, consider a sequence of nonnegative increasing to a limit . Consider setting , which is optional. For a stopping time , monotone convergence gives
Proof of Theorem 2: This follows by an identical proof to the one just given for Theorem 1, with `optional’ replaced by `predictable’, `stopping time’ replaced by `predictable stopping time’, and replaced by . ⬜
The approach to the projection theorems in this post is fairly standard, although there are some slight variations within the literature. For one thing, unlike many texts, I am not assuming that the usual conditions are satisfied. In particular, the filtration need not be right-continuous. The only complication that this leads to is that, in Lemma 7 above, restricting to cadlag modifications of the martingale M is not sufficient, and it is necessary to allow it to be non-right-continuous at a countable set of times.
Some texts on the subject restrict attention to bounded processes. I do not do this. Instead, for more generality, I use weak integrability requirements which are just sufficient to ensure that a real-valued projection exists. This approach is also used, for example, in the book Semimartingale Theory and Stochastic Calculus, 1992, by Shang-wu He, Jia-gang Wang and Jia-an Yan. In the other direction, for non-negative processes, the projections are often defined without any integrability requirements at all. This is because conditional expectations can be defined for all non-negative random variables, so equations (1) and (2) are well-defined for non-negative processes X. However, in this case, the projections can be infinite.