Advertisement

The Doob–Meyer Decomposition

  • Samuel N. Cohen
  • Robert J. Elliott
Part of the Probability and Its Applications book series (PA)

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 XY 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

Keywords

Local Time Extension Theorem Previous Chapter Predictable Process Lebesgue Measure Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Samuel N. Cohen
    • 1
  • Robert J. Elliott
    • 2
    • 3
  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.School of MathematicsUniversity of AdelaideAdelaideAustralia
  3. 3.Haskayne School of BusinessUniversity of CalgaryCalgaryCanada

Personalised recommendations