Abstract
We recall classical facts from the theory of stochastic processes, which will play an important role in the following chapters. We pay special attention to projections and dual projections which are essential for the study of random times. References for the recalled results are given in bibliographic notes at the end of the chapter.
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- 1.
In French, right-continuous is continu à droite, and with left-limits is admettant des limites à gauche. We shall also use càd for right-continuous and càglàd for left-continuous with right-limits. The use of these acronyms comes from P.-A. Meyer.
- 2.
H comes from Hardy.
- 3.
Class (D) is in honor of Doob.
- 4.
In other terms, \(V^p\) is the compensator of V, with \(V^p_{0-}=0\).
- 5.
A process X is a sigma-martingale if there exists a martingale Y and an Y-integrable predictable non-negative process \(\varphi \) such that \(X=\varphi \varvec{\cdot }Y\). A local martingale is a sigma-martingale.
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Aksamit, A., Jeanblanc, M. (2017). Stochastic Processes. In: Enlargement of Filtration with Finance in View. SpringerBriefs in Quantitative Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-41255-9_1
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DOI: https://doi.org/10.1007/978-3-319-41255-9_1
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