Expand knowledge base

1 files changed, 29 insertions, 1 deletions

diff --git a/content/know/concept/sigma-algebra/index.pdc b/content/know/concept/sigma-algebra/index.pdc
index 690c4cc..1a459ea 100644
--- a/content/know/concept/sigma-algebra/index.pdc
+++ b/content/know/concept/sigma-algebra/index.pdc
@@ -43,7 +43,7 @@ is a sub-family of a certain $\mathcal{F}$,
 which is a valid $\sigma$-algebra in its own right.
 
 
-## Notable examples
+## Notable applications
 
 A notable $\sigma$-algebra is the **Borel algebra** $\mathcal{B}(\Omega)$,
 which is defined when $\Omega$ is a metric space,
@@ -54,6 +54,8 @@ and all the subsets of $\mathbb{R}$ obtained by countable sequences
 of unions and intersections of those intervals.
 The elements of $\mathcal{B}$ are **Borel sets**.
 
+<hr>
+
 Another example of a $\sigma$-algebra is the **information**
 obtained by observing a [random variable](/know/concept/random-variable/) $X$.
 Let $\sigma(X)$ be the information generated by observing $X$,
@@ -84,6 +86,32 @@ if $Y$ can always  be computed from $X$,
 i.e. there exists a function $f$ such that
 $Y(\omega) = f(X(\omega))$ for all $\omega \in \Omega$.
 
+<hr>
+
+The concept of information can be extended for
+stochastic processes (i.e. time-indexed random variables):
+if $\{ X_t : t \ge 0 \}$ is a stochastic process,
+its **filtration** $\mathcal{F}_t$ contains all
+the information generated by $X_t$ up to the current time $t$:
+
+$$\begin{aligned}
+    \mathcal{F}_t
+    = \sigma(X_s : 0 \le s \le t)
+\end{aligned}$$
+
+In other words, $\mathcal{F}_t$ is the "accumulated" $\sigma$-algebra
+of all information extractable from $X_t$,
+and hence grows with time: $\mathcal{F}_s \subset \mathcal{F}_t$ for $s < t$.
+Given $\mathcal{F}_t$, all values $X_s$ for $s \le t$ can be computed,
+i.e. if you know $\mathcal{F}_t$, then the present and past of $X_t$ can be reconstructed.
+
+Given some filtration $\mathcal{H}_t$, a stochastic process $X_t$
+is said to be *"$\mathcal{H}_t$-adapted"*
+if $X_t$'s own filtration  $\sigma(X_s : 0 \le s \le t) \subseteq \mathcal{H}_t$,
+meaning $\mathcal{H}_t$ contains enough information
+to determine the current and past values of $X_t$.
+Clearly, $X_t$ is always adapted to its own filtration.
+
 
 
 ## References