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Erschienen in:
2016 | OriginalPaper | Buchkapitel
3. Filtrations and Martingales
verfasst von : Jean-François Le Gall
Erschienen in: Brownian Motion, Martingales, and Stochastic Calculus
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Abstract
In this chapter, we provide a short introduction to the theory of continuous time random processes on a filtered probability space. On the way, we generalize several notions introduced in the previous chapter in the framework of Brownian motion, and we provide a thorough discussion of stopping times. In a second step, we develop the theory of continuous time martingales, and, in particular, we derive regularity results for sample paths of martingales. We finally discuss the optional stopping theorem for martingales and supermartingales, and we give applications to explicit calculations of distributions related to Brownian motion.