Definition:Natural Filtration/Continuous Time

Definition
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\sequence {X_t}_{t \ge 0}$ be an $\hointr 0 \infty$-indexed family of real-valued random variables.

We define the natural filtration $\sequence {\FF_t^X}_{t \ge 0}$ by:


 * $\FF_t^X = \map \sigma {X_s : s \le t}$

for each $t \in \hointr 0 \infty$, where $\map \sigma {X_s : s \le t}$ is the $\sigma$-algebra generated by the family $\set {X_s : s \le t}$.