The Progressive, Optional and Predictable σ-Algebras

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.