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Published in:
2015 | OriginalPaper | Chapter
7. The Progressive, Optional and Predictable σ-Algebras
Authors : Samuel N. Cohen, Robert J. Elliott
Published in: Stochastic Calculus and Applications
Publisher: Springer New York
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Abstract
We now move from looking at different types of stopping times to different types of processes. Recall that we defined a real-valued process Y to be progressive (Definition 3.2.25) if, for every t, the map (s, ω) ↦ X
s
(ω) of [0, t] ×Ω into \((\mathbb{R},\mathcal{B}(\mathbb{R}))\) is measurable, when [0, t] ×Ω is given the product σ-algebra \(\mathcal{B}([0,t]) \otimes \mathcal{F}_{t}\). Essentially, this states that the process X is adapted and is Borel measurable with respect to time.