Definition:Filtration of Sigma-Algebra/Continuous Time

Definition

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

Let $\sequence {\FF_t}_{t \ge 0}$ be an $\hointr 0 \infty$-indexed family of sub-$\sigma$-algebras of $\Sigma$ such that:

$\FF_t \subseteq \FF_s$ whenever $t, s \in \hointr 0 \infty$ are such that $t \le s$.

We say that $\sequence {\FF_t}_{t \ge 0}$ is a filtration of $\Sigma$.

Sources

• 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): Definition $3.1$