Filtered sigma algebra

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In mathematics, a filtered σ-algebra is a σ-algebra on some space $X$ , along with a filtration of that σ-algebra with respect to some index set.

Similarly, a filtered probability space (also known as a stochastic basis) is a probability space with a filtration of its σ-algebra. Filtered probability spaces are often used in the study of stochastic processes: for example, canonical Brownian motion $B$ on the real line $\mathbb{R}$ would be studied using the filtered probability space

$\left( C_{0}, \mathcal{F}, \mathcal{F}_{*}^{B}, \gamma \right),$

where

$C 0$ denotes classical Wiener space;
$\mathcal{F}$ denotes the Borel σ-algebra on $C 0$ ;
$\mathcal{F}_{*}^{B}$ denotes the natural filtration of $\mathcal{F}$ by $B$ :

$\mathcal{F}_{t}^{B} := \sigma \{ B_{s}^{-1} (A) | A \mbox{ is a Borel subset of } \mathbb{R} \mbox{ and } 0 \leq s \leq t \} \subseteq \mathcal{F};$

$γ$ denotes classical Wiener measure.

[edit] Reference

Shiryaev, A.N., "Stochastic integral" SpringerLink Encyclopaedia of Mathematics (2001)

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Categories: Measure theory | Stochastic processes

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