Martingale decompositions and integration

  • M. M. Rao
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 342)

Abstract

Continuing the work of the preceding chapter on continuous parameter (sub)martingales, we present a solution of the Doob decomposition problem, raised in Section 2.5, which is due to Meyer. We give an elementary (but longer) demonstration and also sketch a shorter (but more sophisticated) argument based on Doléans-Dade signed measure representation of quasimartingales. This decomposition leads to stochastic integration with square integrable martingales as integrators generalizing the classical Itô integration. The material is presented in this chapter in considerable detail, since it forms a basis for semimartingale integrals with numerous applications to be abstracted and treated in the following chapter. Orthogonal decompositions of square integrable martingales (of continuous time parameter), its time change transformation leading to a related Brownian motion process and the Lévy characterization of the latter from continuous parameter martingales, are covered. Stopping (or optional) times play a key role in all this work, and some classifications of these are given. The treatment also includes the Stratonovich integrals as well as an identification of the square integrable martingale integrators with spectral measures of certain normal operators in Hilbert space. Finally some related results appear as exercises in the Complements section.

Keywords

Filtration Manifold Covariance limE Radon 

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.University of CaliforniaRiversideUSA

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