Martingale Convergence Theorems and Their Applications

  • Achim Klenke
Part of the Universitext book series (UTX)

Abstract

We became familiar with martingales X=(X n ) nN0 as fair games and found that under certain transformations (optional stopping, discrete stochastic integral) martingales turn into martingales. In this chapter, we will see that under weak conditions (non-negativity or uniform integrability) martingales converge almost surely. Furthermore, the martingale structure implies L p -convergence under assumptions that are (formally) weaker than those of Chapter  7. The basic ideas of this chapter are Doob’s inequality (Theorem 11.4) and the upcrossing inequality (Lemma 11.3).

Keywords

Filtration 

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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Achim Klenke
    • 1
  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany

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