From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 27 Oct 2022 20:40:09 +0200
Subject: Optimize last images, add proof template, improve CSS
---
source/know/concept/ito-integral/index.md | 24 +++++++++---------------
1 file changed, 9 insertions(+), 15 deletions(-)
(limited to 'source/know/concept/ito-integral')
diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md
index f087f97..4a725e1 100644
--- a/source/know/concept/ito-integral/index.md
+++ b/source/know/concept/ito-integral/index.md
@@ -29,6 +29,7 @@ and $$\mathbf{E}[G_t^2]$$ is integrable for $$t \in [a, b]$$.
If $$I_t$$ exists, $$G_t$$ is said to be **Itō-integrable** with respect to $$B_t$$.
+
## Motivation
Consider the following simple first-order differential equation for $$X_t$$,
@@ -99,7 +100,8 @@ $$\begin{aligned}
\end{aligned}$$
For more information about applying the Itō integral in this way,
-see the [Itō calculus](/know/concept/ito-calculus/).
+see the [Itō calculus](/know/concept/ito-process/).
+
## Properties
@@ -138,11 +140,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-isometry"/>
-<label for="proof-isometry">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-isometry">Proof.</label>
+
+{% include proof/start.html id="proof-isometry" -%}
We write out the left-hand side of the Itō isometry,
where eventually $$h \to 0$$:
@@ -208,20 +207,16 @@ $$\begin{aligned}
\longrightarrow
\int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t}
\end{aligned}$$
+{% include proof/end.html id="proof-isometry" %}
-</div>
-</div>
Furthermore, Itō integrals are [martingales](/know/concept/martingale/),
meaning that the average noise contribution is zero,
which makes intuitive sense,
since true white noise cannot be biased.
-<div class="accordion">
-<input type="checkbox" id="proof-martingale"/>
-<label for="proof-martingale">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-martingale">Proof.</label>
+
+{% include proof/start.html id="proof-martingale" -%}
We will prove that an arbitrary Itō integral $$I_t$$ is a martingale.
Using additivity, we know that the increment $$I_t \!-\! I_s$$
is as follows, given information $$\mathcal{F}_s$$:
@@ -259,8 +254,7 @@ so from the Itō isometry we have $$\mathbf{E}[I]^2 < \infty$$,
and therefore $$\mathbf{E}[I] < \infty$$,
so $$I_t$$ has all the properties of a Martingale,
since it is trivially $$\mathcal{F}_t$$-adapted.
-</div>
-</div>
+{% include proof/end.html id="proof-martingale" %}
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