Filtration

Definition

A filtration $\mathcal{F}$ is an indexed family $(S_i{)}_{i\in I}$ of subobjects of a given Algebraic Structure $S$, with the index $i$ running over some Totally Ordered Set $I$, subject to the condition that $i\le j\in I\Rightarrow S_i\subset S_j$.

Examples

Measure Theory

In measure theory, a filtration is an increasing Sequence of Sigma-Algebra on a Measurable Space. Given a Measurable Space $(\Omega ,\mathcal{F})$, a filtration is a sequence of sub-$\sigma$-algebras $\{{\mathcal{F}}_i{\}}_{t\ge 0}$ with ${\mathcal{F}}_i\subset \mathcal{F}$ where $t\in {\mathbb{R}}_0^{+}$ and $t_1\le t_2\Rightarrow {\mathcal{F}}_{t_1}\subset {\mathcal{F}}_{t_2}$.

A filtered probability space $(\Omega ,\mathcal{F},\{{\mathcal{F}}_i{\}}_{t\ge 0},\mathbb{P})$ is a Probability Space equipped with the filtration $\{{\mathcal{F}}_i{\}}_{t\ge 0}$ of its Sigma-Algebra $\mathcal{F}$.