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Published in:
2016 | OriginalPaper | Chapter
4. Continuous Semimartingales
Author : Jean-François Le Gall
Published in: Brownian Motion, Martingales, and Stochastic Calculus
Publisher: Springer International Publishing
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Abstract
Continuous semimartingales provide the general class of processes with continuous sample paths for which we will develop the theory of stochastic integration in the next chapter. By definition, a continuous semimartingale is the sum of a continuous local martingale and a (continuous) finite variation process. In the present chapter, we study separately these two classes of processes. We start with some preliminaries about deterministic functions with finite variation, before considering the corresponding random processes. We then define (continuous) local martingales and we construct the quadratic variation of a local martingale, which will play a fundamental role in the construction of stochastic integrals. We explain how properties of a local martingale are related to those of its quadratic variation. Finally, we introduce continuous semimartingales and their quadratic variation processes.