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.
9 thoughts on “The General Theory of Semimartingales”
I am really looking forward reading this section of your blog !!!
By the way, do you consider opening a section about Fundamental Asset Pricing Theorems in their most general forms (i.e. Schachermayer and Delbaen forms) ?
Fantastic! This theory is really interesting stuff, so I’m glad you’re looking forward to reading it. Bear with me though – I just decided to re-order the upcoming posts, as there is already enough enough theory in the notes already published to give a fairly quick proof of the Bichteler-Dellacherie theorem. So, I have decided to post that first instead. Hopefully should be able to post this early next week.
And, regarding your suggestion about the Fundamental Theorems of Asset Pricing. That is interesting stuff, and something that I worked on a while ago (very closely related to the results of Delbaen and Schachermayer). I didn’t get round to publishing this though. There’s a very nice duality between the equivalent martingale measures and arbitrage bounds for pricing contingent claims (and the existence or absence of arbitrage). I have thought about writing some posts related to this, and will bear your suggestion in mind, but I can’t promise that I’ll get round to doing this very soon.
Thank’s anyway your blog is really making things that are hard to get, “almost surely” easy to undestand.
Regarding FTAP, I have always thought that the “Schachermayer and Delbaen” proofs were using arguments that were insufficiently detailed (for my level of understanding and/or education), especially regarding the Functional Analysis arguments they use, and their book is insufficiently self-contained in this regard in my opinion. So there is a “gap” that I thought you would be both able and interested to close.
By the way there is a Mathematical-Finance oriented proof of the Bichteler-Dellacherie’s theorem (by Schachermayer, Veliyev and Beiglböck) that you can find here :
PS : there is also Doob-Meyer Theorem proof in the same vein that you cal find on Arxiv
Thanks for that link. It looks very interesting. The proof they give avoids the use of the Hahn-Banach theorem usually used to prove Bichteler-Dellacherie, so it is more “constructive” than the standard proof, and logically closer to the proof that I am about to post here.
Also, apologies for being a bit slow in getting my post on Bichteler-Dellacherie up. I just haven’t had any free time to get this written up over the last week. Finally, I’ve got a bit of time today, so I should be able to finish this off and post it tomorrow.
Check Josef’ Teichmanns notes concerning a concise and simple introduction to general stochastic integration and a simpliefied proof of the FTAP https://people.math.ethz.ch/~jteichma/lecturenotesMF20131220.pdf
And here the simplified proof of the FTAP which relies fundamentally on a result of Kostas Kardaras about generalized supermartingale deflators under limited information. This simplifies in particular the second (and most difficult) part of the original FTAP: https://arxiv.org/abs/1406.5414