Stochastic Calculus and Applications pp 199-210 | Cite as

# The Doob–Meyer Decomposition

## Abstract

In the previous chapter, we have seen that, for any process \(X \in \mathcal{A}\), we can find a predictable process *Y* = *Π*_{ p }^{∗}*X* such that *X* − *Y* is a martingale. This is a fundamentally useful property, and in this chapter we show that a similar decomposition holds for all right-continuous local supermartingales (and hence local submartingales). To obtain this, we first consider the particularly ‘nice’ class of processes given by class (D). Recall that a right-continuous uniformly integrable supermartingale *X* is said to be of class (D) if the set of random variables \(\{X_{T}\}_{T\in \mathcal{T}}\) is uniformly integrable (where \(\mathcal{T}\) is the set of all stopping times). These were introduced in Section 5.6