We move on to properties of stochastic integration which, while being fairly elementary, are rather difficult to prove directly from the definitions.
First, recall that for a semimartingale X, the X-integrable processes were defined to be predictable processes which are `good dominators’. That is, if are bounded predictable processes with and pointwise, then tends to zero in probability. This definition is a bit messy. Fortunately, the following result gives a much cleaner characterization of X-integrability.
Theorem 1 Let X be a semimartingale. Then, a predictable process is X-integrable if and only if the set
is bounded in probability for each .
Proof: That it is necessary for the set in (1) to be bounded in probability follows from dominated convergence. If satisfy and go to zero, then and dominated convergence gives in probability. By the sequential characterization of boundedness, the given set is indeed bounded in probability.
Conversely, suppose that the set in (1) is bounded in probability. It needs to be shown that, for any sequence with and then in probability.
First, suppose that . Choosing any then is bounded by . So, the collection of all sums of the form
Choosing any we can now show, by contradiction, that there is a positive constant L with for all bounded predictable processes . If this was not the case, then for any increasing sequence of numbers there would exist bounded predictable such that
Furthermore, by bounded convergence, (2) still holds if is replaced by for large enough m. Then, by passing to a subsequence if necessary, we may suppose that . So,
so that, by the above argument, tends to zero in probability, contradicting (2).
Now, suppose that L is as above and that tend to zero. By bounded convergence, tends to zero in probability and,
for all . As was arbitrary, this shows that in probability as required. ⬜
Changes of Filtration
Recall that we work with respect to a complete filtered probability space . Sometimes it can be useful to replace by an alternative filtration , while keeping the same underlying probability space . It is assumed that , so changing filtrations in this way gives a new filtered probability space . For example, if studying a process which is not adapted, changing to a new filtration with respect to which it is adapted can be useful technique.
Changing filtrations in this way affects the notion of adapted processes and, therefore, the space of predictable processes changes. As stochastic integration was defined only for predictable integrands, it is also affected by changes of filtration. If a process X is a semimartingale with respect to F, it can fail to be a semimartingale with respect to a different filtration G even in the case where X is also G-adapted and. Even if it is, the space of X-integrable processes and the values of the stochastic integral can differ.
I will say F-predictable, F-semimartingale, etc to denote that the respective property holds with respect to a given filtration F. Also, if X is an F-semimartingale, denotes the X-integrable processes with respect to F.
The filtration G is said to be a subfiltration of F if for each t. This case is particularly easy, because passing to a subfiltration reduces the set of predictable processes, making it easier for there to be a well defined stochastic integral.
Theorem 2 (Stricker’s Theorem) Let X be a semimartingale for the filtration F, and G be a subfiltration of F such that X is G-adapted. Then, X is a G-semimartingale.
Furthermore, if then the integrals defined with respect to F and G agree.
Proof: As G is a subfiltration of F, every G-predictable process is also F-predictable. We can therefore define the integral for bounded G-predictable to be equivalent to the value defined with respect to F. This agrees with the explicit expression for G-elementary processes and satisfies bounded convergence in probability.
So, X is a G-semimartingale by definition and the stochastic integral defined under G for bounded integrands agrees with the definition under F. Now, suppose that . Then, dominated convergence
applies under both filtrations, and the two definitions of the integral agree. ⬜
Going in the opposite direction and increasing the size of the filtration makes the set of predictable processes larger, so it becomes less likely that there is a well-defined stochastic integral. For example, a standard Brownian motion is a semimartingale under its natural filtration but, as it has infinite variation on bounded sets, it is not a semimartingale under the maximal filtration (where for all t).
However, enlarging the filtration by adding a single set to , the semimartingale property is preserved. In fact, enlarging the filtration by any countable collection of disjoint sets preserves the semimartingale property. This is known as Jacod’s Countable Expansion. Furthermore, X-integrability is also preserved.
Theorem 3 Let be a sequence of pairwise disjoint sets in and, for each t, let be the sigma algebra generated by . So, is a filtration containing F.
Then, every F-semimartingale X is also a G-semimartingale. Furthermore, and, for , the definitions of the integral with respect to F and G agree.
Proof: First, by inserting into the sequence of sets, we can suppose that . I make use of the result that a set S of random variables is bounded in probability if and only if is bounded in probability for each n.
The enlarged filtration G and its associated predictable processes are not hard to describe. First, is the collection of sets of the form for . The -measurable random variables are of the form for -measurable random variables . Then, the G-elementary processes are for F-elementary processes . Similarly, the G-predictable processes are of the form for F-predictable .
The characterization of semimartingales in terms of boundedness in probability will be used. Suppose that X is an F-semimartingale and, for each t, define the sets
For any G-elementary process , there are F-elementary processes with . Replacing by , we suppose that . Then, . So, .
The following implications hold. X is an F-semimartingale implies that S is bounded in probability, implying that are bounded in probability, so are bounded in probability. Then, is bounded in probability and X is a G-semimartingale.
It only remains to prove that , in which case the agreement of stochastic integration defined with respect to F and G comes from Theorem 2.
The proof that X-integrability is preserved when enlarging the filtration follows in a similar way as for the semimartingale property above. The characterization of X-integrability in terms of boundedness in probability given by Theorem 1 is used. So, suppose that and define
For any bounded G-predictable process , there are bounded F-predictable such that . Replacing by , we suppose that . Then, , proving that .
In a similar way to the proof above, is bounded in probability implies are bounded in probability, implying that are bounded in probability, so is bounded in probability and . ⬜
Finally, for this post, we prove the following `pathwise’ property of stochastic integration. The stochastic integrals with respect to two different integrands agree on any measurable set for which the integrands agree. If integration was defined in a pathwise manner, such as when taking the standard Stieltjes integrals with respect to finite variation processes, then this result would be immediate. However, it appears to be very difficult to prove from the defining properties of stochastic integration. Instead, Theorem 3 above is used which, in turn, relied on the alternative definition of semimartingales in terms of boundedness in probability.
Theorem 4 Let X be a semimartingale and . If is any set such that on A, then on A.
Proof: Using Theorem 3, enlarge the filtration by adding the set A to each . Working with respect to this larger filtration, are X-integrable and,
as required. ⬜
The characterization of X-integrable processes given by Theorem 1 is a bit different those used in most introductions to stochastic calculus. The usual approach is to decompose the semimartingale X=M+V into a local martingale part M and FV term V. Then, a process is X-integrable if it is both V-integrable in the Lebesgue-Stieltjes sense and M-integrable according to the construction of stochastic integrals with respect to local martingales. However, such decompositions are not unique, and different decompositions lead to different sets of integrands. Then, a process is said to be X-integrable if it is integrable with respect to at least one such decomposition. It can be shown that this does give the same class of integrands as in Theorem 1, but it is not easy to prove this.
On the other hand, Theorem 1 gives what seems to be a more natural and intrinsic definition of X-integrability. I’m not sure if the result that this is a necessary and sufficient condition is new, and am not aware of any authors using this characterization. It does seem surprising if this is the case, as it is such a simple definition of X-integrable processes.