Abstract
In this chapter, we consider studies involving càdlàg processes. We pay particular attention to semi-martingales with respect to a given filtration; these processes will always be taken with càdlàg paths. We present the definition of stochastic integrals with respect to a square integrable martingale, and we extend the definition to stochastic integrals with respect to a local martingale. Then, we introduce semi-martingales, quadratic covariation processes for semi-martingales and some general versions of Itô’s formula and Girsanov’s theorem. We give necessary and sufficient conditions for the existence of an equivalent martingale measure. We end the chapter with a brief survey of valuation in an incomplete market.
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© 2009 Springer-Verlag London
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Jeanblanc, M., Yor, M., Chesney, M. (2009). General Processes: Mathematical Facts. In: Mathematical Methods for Financial Markets. Springer Finance. Springer, London. https://doi.org/10.1007/978-1-84628-737-4_9
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DOI: https://doi.org/10.1007/978-1-84628-737-4_9
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Publisher Name: Springer, London
Print ISBN: 978-1-84882-819-3
Online ISBN: 978-1-84628-737-4
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