Definition:Natural Filtration/Discrete Time

Definition

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

Let $\sequence {X_n}_{n \ge 0}$ be a sequence of real-valued random variables.

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

$\FF_n^X = \map \sigma {X_0, X_1, \ldots, X_n}$

for each $n \ge 0$, where $\map \sigma {X_0, X_1, \ldots, X_n}$ is the $\sigma$-algebra generated by $\sequence {X_0, X_1, \ldots, X_n}$.

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $10.1$: Filtered Spaces