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,