As previously discussed, for discrete-time processes the Doob decomposition is a simple, but very useful, technique which allows us to decompose any integrable process into the sum of a martingale and a predictable process. If is an integrable discrete-time process adapted to a filtration , then the Doob decomposition expresses X as
Then, M is then a martingale and A is an integrable process which is also predictable, in the sense that is -measurable for each . The expected value of the variation of A can be computed in terms of X,
This is the mean variation of X.
In continuous time, the situation is rather more complex, and will require constraints on the process X other than just integrability. We have already discussed the case for submartingales — the Doob-Meyer decomposition. This decomposes a submartingale into a local martingale and a predictable increasing process.
A natural setting for further generalising the Doob-Meyer decomposition is that of quasimartingales. In continuous time, the appropriate class of processes to use for the component A of the decomposition is the predictable FV processes. Decomposition (2) below is the same as that in the previous post on special semimartingales. This is not surprising, as we have already seen that the class of special semimartingales is identical to the class of local quasimartingales. The difference with the current setting is that we can express the expected variation of A in terms of the mean variation of X, and obtain a necessary and sufficient condition for the local martingale component to be a proper martingale.
As was noted in an earlier post, historically, decomposition (2) for quasimartingales played an important part in the development of stochastic calculus and, in particular, in the proof of the Bichteler-Dellacherie theorem. That is not the case in these notes, however, as we have already proven the main results without requiring quasimartingales. As always, any two processes are identified whenever they are equivalent up to evanescence.
Theorem 1 Every cadlag quasimartingale X uniquely decomposes as
so that, in particular,
Furthermore, the following are equivalent,
Proof: We will proceed by extending the Doob-Meyer decomposition for submartingales to the quasimartingale case. So, apply Rao’s decomposition,
Now apply the Doob-Meyer decomposition to each of Y, Z
Here, U,V are local martingales and B,C are integrable increasing predictable processes starting from 0. Decomposition (2) is then given by setting and . The variation of A is bounded by the sum of B and C, so is integrable over each finite time interval.
By the martingale property for , is equal to for any times . So, evaluating along partitions of gives
The final limit is using dominated convergence. Also, using the martingale property of we have that equals . Hence,
It just remains to prove the equivalence of the three statements in the theorem. First, on each finite interval , the process A is bounded by its variation, which is integrable. So, A is always of class (DL). Then, X is of class (DL) if and only if M is. As a local martingale is a proper martingale if and only if it is of class (DL), we see that statements 1 and 2 are equivalent.
The proof above of existence of the decomposition (2) assumed that the underlying filtration is right-continuous. We now show that this condition can be removed. Let be an elementary process with respect to the right-continuous filtration . Then, the process is elementary with respect to the original filtration. By Lemma 5 of the previous post, in as so,
Taking the supremum over all such shows that the mean variation of X over taken with respect to the right-continuous filtration is bounded by , so is finite. We can therefore apply the above proof to obtain decomposition (2) with respect to this filtration. That is, M is a local martingale and A is predictable with respect to .
Next, as and generate the same predictable sigma-algebra, A is a predictable FV process w.r.t. . Now, choose stopping times
Then, the stopped process is bounded by the integrable random variable over the interval . Hence, it is class (DL), so is a proper martingale w.r.t and, hence, is also a martingale w.r.t. the original filtration. So M is a local martingale w.r.t. the original filtration, as required. ⬜
Approximating the Compensator
Finally, we can show that the Doob-Meyer decomposition (2) is indeed the continuous-time limit of the discrete-time Doob decomposition (1). The remainder of this post closely follows the argument given in the previous posts on compensators and the Doob-Meyer submartingale decomposition.
We discretize time using a stochastic partition P of , which is a sequence of stopping times
The mesh of the partition is . The compensator, , of X computed along the partition P is
We will consider class (D) processes X for which is finite. The class (D) property ensures that X is -bounded, so
is finite. By quasimartingale convergence, the limit exists, and is integrable for all stopping times . So, the expectations in (6) are well defined. Theorem 1 says that for a martingale M and predictable FV process A with integrable variation,
This ensures that exists and is integrable, and A is of class (D). Therefore, M is a class (D) martingale. Optional sampling implies that , so (6) can be rewritten to express in terms of A,
In the case where X is quasi-left-continuous, so that A is continuous, the approximation to the compensator calculated along partitions P converges uniformly in to A as the mesh goes to zero. The notation denotes the limit as the mesh goes to zero in probability.
Theorem 2 Let X be a cadlag and quasi-left-continuous process of class (D), with finite mean variation . Then, with A as in decomposition (2), uniformly in as in probability. That is,
If the quasimartingale X is not quasi-left-continuous, then the discrete-time compensator is not guaranteed to converge in . An explicit example where this is the case was given in the post on compensators of stopping times. Instead, we have to work with respect to weak convergence in .
Theorem 3 Let X be a cadlag class (D) process with finite mean variation, . Then, with A as in decomposition (2), weakly in as in probability, for any random time . That is,
for all uniformly bounded random variables Y.