Definition:Natural Filtration/Continuous Time
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Definition
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 $\set {X_s : s \le t}$.
Sources
- 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): $3.1$: Filtrations and Processes