Having completed the series of posts applying the methods of stochastic calculus to various special types of processes, I now return to the development of the theory. The next few posts of these notes will be grouped under the heading `The General Theory of Semimartingales’. Subjects which will be covered include the classification of predictable stopping times, integration with respect to continuous and predictable FV processes, decompositions of special semimartingales, the Bichteler-Dellacherie theorem, the Doob-Meyer decomposition and the theory of quasimartingales.
One of the main results is the Bichteler-Dellacherie theorem describing the class of semimartingales which, in these notes, were defined to be cadlag adapted processes with respect to which the stochastic integral can be defined (that is, they are good integrators). It was shown that these include the sums of local martingales and FV processes. The Bichteler-Dellacherie theorem says that this is the full class of semimartingales. Classically, semimartingales were defined as a sum of a local martingale and an FV process so, an alternative statement of the theorem is that the classical definition agrees with the one used in these notes. Further results, such as the Doob-Meyer decomposition for submartingales and Rao’s decomposition for quasimartingales, will follow quickly from this.
Logically, the structure of these notes will be almost directly opposite to the historical development of the results. Originally, much of the development of the stochastic integral was based on the Doob-Meyer decomposition which, in turn, relied on some advanced ideas such as the predictable and dual predictable projection theorems. However, here, we have already introduced stochastic integration without recourse to such general theory, and can instead make use of this in the theory. The reasons I have taken this approach are as follows. First, stochastic integration is a particularly straightforward and useful technique for many applications, so it is desirable to introduce this early on. Second, although it is possible to use the general theory of processes in the construction of the integral, such an approach seems rather distinct from the intuitive understanding of stochastic integration as well as superfluous to many of its properties. So it seemed more natural from the point of view of these notes to define the integral first, guided by the properties of the (non-stochastic) Lebesgue integral, then show how its elementary properties follow from the definitions, and develop the further theory later.
In particular, the proof I will give of the Bichteler-Dellacherie theorem is very different from the standard method. A standard way of proving this result goes roughly as follows. Given a process satisfying the necessary conditions show that, under an equivalent change of measure, the process becomes a quasimartingale. This measure change is rather arbitrary, requiring an application of the Hahn-Banach theorem. Rao’s theorem, which is an extension of the Doob-Meyer decomposition to quasimartingales, gives the required decomposition. Finally, it is shown that the decomposition still exists after changing back to the original measure. On the other hand, the method used here only requires working under the original measure. As the processes under consideration have well-defined quadratic variations and covariations, we can use this to decompose the space of semimartingales into the local martingales plus an `orthogonal complement’. This complement consists of the semimartingales such that is a local martingale for all martingales (and, in the continuous case, is simply the set of semimartingales with zero quadratic variation). The proof of the Bichteler-Dellacherie theorem and related semimartingale decompositions reduces to showing that this orthogonal complement consists precisely of the predictable FV processes.