From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/stochastic-process/index.md | 48 ++++++++++++-------------
 1 file changed, 24 insertions(+), 24 deletions(-)

(limited to 'source/know/concept/stochastic-process')

diff --git a/source/know/concept/stochastic-process/index.md b/source/know/concept/stochastic-process/index.md
index dc3d30e..68f028d 100644
--- a/source/know/concept/stochastic-process/index.md
+++ b/source/know/concept/stochastic-process/index.md
@@ -9,24 +9,24 @@ categories:
 layout: "concept"
 ---
 
-A **stochastic process** $X_t$ is a time-indexed
+A **stochastic process** $$X_t$$ is a time-indexed
 [random variable](/know/concept/random-variable/),
-$\{ X_t : t > 0 \}$, i.e. a set of (usually correlated)
-random variables, each labelled with a unique timestamp $t$.
+$$\{ X_t : t > 0 \}$$, i.e. a set of (usually correlated)
+random variables, each labelled with a unique timestamp $$t$$.
 
 Whereas "ordinary" random variables are defined on
-a probability space $(\Omega, \mathcal{F}, P)$,
+a probability space $$(\Omega, \mathcal{F}, P)$$,
 stochastic process are defined on
-a **filtered probability space** $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P)$.
-As before, $\Omega$ is the sample space,
-$\mathcal{F}$ is the event space,
-and $P$ is the probability measure.
+a **filtered probability space** $$(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P)$$.
+As before, $$\Omega$$ is the sample space,
+$$\mathcal{F}$$ is the event space,
+and $$P$$ is the probability measure.
 
-The **filtration** $\{ \mathcal{F}_t : t \ge 0 \}$
-is a time-indexed set of [$\sigma$-algebras](/know/concept/sigma-algebra/) on $\Omega$,
+The **filtration** $$\{ \mathcal{F}_t : t \ge 0 \}$$
+is a time-indexed set of [$$\sigma$$-algebras](/know/concept/sigma-algebra/) on $$\Omega$$,
 which contains at least all the information generated
-by $X_t$ up to the current time $t$,
-and is a subset of $\mathcal{F}_t$:
+by $$X_t$$ up to the current time $$t$$,
+and is a subset of $$\mathcal{F}_t$$:
 
 $$\begin{aligned}
     \mathcal{F}
@@ -34,18 +34,18 @@ $$\begin{aligned}
     \supseteq \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: $\mathcal{F}_s \subseteq \mathcal{F}_t$ 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 any 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.
+In other words, $$\mathcal{F}_t$$ is the "accumulated" $$\sigma$$-algebra
+of all information extractable from $$X_t$$,
+and hence grows with time: $$\mathcal{F}_s \subseteq \mathcal{F}_t$$ 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 any 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.
 
 Filtration and their adaptations are very useful
 for working with stochastic processes,
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