ABSTRACT
One of the most fundamental theorems in the martingale theory is the Doob-Meyer decomposition theorem. It says that any submartingale can be uniquely decomposed into the sum of a predictable, increasing process and a martingale. Although this claim may not look exciting or interesting at first sight, the true worth of the theorem is that the decomposition is unique. This point is closely related to the fact, which is easy to remember, that any (local) martingale starting from zero which is “predictable” and has “finite-variation” (on each compact interval) is necessarily zero (the degenerate process).
This explanation has been chosen as an introduction to this chapter in order to announce at this stage that two of the important concepts in the martingale theory are “predictability” and “finite-variation”. With these two properties in hands, we will be able to build up a lot of important objects in the theory, including the “predictable quadratic (co)-variation” of a square-integrable martingale.
The chapter finishes with an explanation about a deep theory concerning the decomposition of local martingales.
