The Doob–Meyer Decomposition

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


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


Local Time Extension Theorem Previous Chapter Predictable Process Lebesgue Measure Zero 
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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

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