Definition
Filtration and Adapted Process
A filtration on $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~Probability Space) is an increasing family $(\mathcal{F}_t)_{t \geq 0}$ of sub-$\sigma$-algebras of $\mathcal{F}$:
\[
s \leq t \implies \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}.
\]
A filtration is right-continuous if $\mathcal{F}_t = \bigcap_{s > t} \mathcal{F}_s$ for all $t$.
A process $\{X_t\}$ (Definition~Stochastic Process) is adapted to $(\mathcal{F}_t)$ if $X_t$ is $\mathcal{F}_t$-measurable for every $t \geq 0$.
The natural filtration of $\{X_t\}$ is $\mathcal{F}_t^X := \sigma(X_s : s \leq t)$.
Depends on
Used in
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
nbbb6ebd1["Probability Space"]
nccabff62["Stochastic Process"]
ne6452d7c["Filtration and Adapted Process"]:::current
n96541dd7["Martingale"]
nedbd148e["Stopping Time"]
nbbb6ebd1 --> ne6452d7c
nccabff62 --> ne6452d7c
ne6452d7c --> n96541dd7
ne6452d7c --> nedbd148e
click nbbb6ebd1 "../objects/bbb6ebd1.html" "_self"
click nccabff62 "../objects/ccabff62.html" "_self"
click n96541dd7 "../objects/96541dd7.html" "_self"
click nedbd148e "../objects/edbd148e.html" "_self"