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2014 | OriginalPaper | Buchkapitel
11. Martingale Convergence Theorems and Their Applications
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Abstract
We became familiar with martingales X=(X
n
)
n∈N0 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).