For stochastic processes in discrete time, the Doob decomposition uniquely decomposes any integrable process into the sum of a martingale and a predictable process. If is an integrable process adapted to a filtration then we write . Here, M is a martingale, so that , and A is predictable with . By saying that A is predictable, we mean that is measurable for each . It can be seen that this implies that
Then it is possible to write A and M as
So, the Doob decomposition is unique and, conversely, the processes A and M constructed according to equation (1) can be seen to be respectively, a predictable process starting from zero and a martingale. For many purposes, this allows us to reduce problems concerning processes in discrete time to simpler statements about martingales and separately about predictable processes. In the case where X is a submartingale then things reduce further as, in this case, A will be an increasing process.
The situation is considerably more complicated when looking at processes in continuous time. The extension of the Doob decomposition to continuous time processes, known as the Doob-Meyer decomposition, was an important result historically in the development of stochastic calculus. First, we would usually restrict attention to sufficiently nice modifications of the processes and, in particular, suppose that X is cadlag. When attempting an analogous decomposition to the one above, it is not immediately clear what should be meant by the predictable component. The continuous time predictable processes are defined to be the set of all processes which are measurable with respect to the predictable sigma algebra, which is the sigma algebra generated by the space of processes which are adapted and continuous (or, equivalently, left-continuous). In particular, all continuous and adapted processes are predictable but, due to the existence of continuous martingales such as Brownian motion, this means that decompositions as sums of martingales and predictable processes are not unique. It is therefore necessary to impose further conditions on the term A in the decomposition. It turns out that we obtain unique decompositions if, in addition to being predictable, A is required to be cadlag with locally finite variation (an FV process). The processes which can be decomposed into a local martingale and a predictable FV process are known as special semimartingales. This is precisely the space of locally integrable semimartingales. As usual, we work with respect to a complete filtered probability space and two stochastic processes are considered to be the same if they are equivalent up to evanescence.
Theorem 1 For a process X, the following are equivalent.
- X is a locally integrable semimartingale.
- X decomposes as
for a local martingale M and predictable FV process A.
Furthermore, choosing , decomposition (2) is unique.
Theorem 1 is a general version of the Doob-Meyer decomposition. However, the name `Doob-Meyer decomposition’ is often used to specifically refer to the important special case where X is a submartingale. Historically, the theorem was first stated and proved for that case, and I will look at the decomposition for submartingales in more detail in a later post. Continue reading “Special Semimartingales”