Category:Definitions/Natural Filtrations

This category contains definitions related to Natural Filtrations.
Related results can be found in Category:Natural Filtrations.

Discrete Time

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 $\sequence {X_0, X_1, \ldots, X_n}$.

Continuous Time

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 $\set {X_s : s \le t}$.