From 075683cdf4588fe16f41d9f7b46b9720b42b2553 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 17 Jul 2024 10:01:43 +0200
Subject: Improve knowledge base

---
 source/know/concept/ito-integral/index.md | 57 +++++++++++++------------------
 1 file changed, 24 insertions(+), 33 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 4a725e1..9b092d6 100644
--- a/source/know/concept/ito-integral/index.md
+++ b/source/know/concept/ito-integral/index.md
@@ -10,8 +10,7 @@ layout: "concept"
 
 The **Itō integral** offers a way to integrate
 a given [stochastic process](/know/concept/stochastic-process/) $$G_t$$
-with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$,
-which is also a stochastic process.
+with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$.
 The Itō integral $$I_t$$ of $$G_t$$ is defined as follows:
 
 $$\begin{aligned}
@@ -47,21 +46,21 @@ which can be applied recursively, leading to:
 $$\begin{aligned}
     X_{t+h}
     \approx X_{t} + f(X_t) \: h
-    \quad \implies \quad
+    \qquad \implies \qquad
     X_t
     \approx X_0 + \sum_{s = 0}^{s = t} f(X_s) \: h
 \end{aligned}$$
 
-In the limit $$h \to 0$$, this leads to the following unsurprising integral for $$X_t$$:
+In the limit $$h \to 0$$, this unsurprisingly leads to the following integral for $$X_t$$:
 
 $$\begin{aligned}
-    \int_0^t f(X_s) \dd{s}
-    = \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h
+    \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h
+    = \int_0^t f(X_s) \dd{s}
 \end{aligned}$$
 
 In contrast, consider the *stochastic differential equation* below,
 where $$\xi_t$$ represents white noise,
-which is informally the $$t$$-derivative
+which is informally defined as the $$t$$-derivative
 of the Wiener process $$\xi_t = \idv{B_t}{t}$$:
 
 $$\begin{aligned}
@@ -89,9 +88,9 @@ $$\begin{aligned}
     = X_0 + \int_0^t g(X_s) \dd{B_s}
 \end{aligned}$$
 
-This integral is *defined* as below,
-analogously to the first, but with $$h$$ replaced by
-the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process.
+The meaning of such an integral is *defined* below.
+It is analogous to the deterministic case,
+but $$h$$ is replaced by the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process.
 This is an Itō integral:
 
 $$\begin{aligned}
@@ -100,7 +99,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 For more information about applying the Itō integral in this way,
-see the [Itō calculus](/know/concept/ito-process/).
+see [Itō calculus](/know/concept/ito-process/).
 
 
 
@@ -131,7 +130,7 @@ $$\begin{aligned}
 A more interesting property is the **Itō isometry**,
 which expresses the expectation of the square of an Itō integral of $$G_t$$
 as a simpler "ordinary" integral of the expectation of $$G_t^2$$
-(which exists by the definition of Itō-integrability):
+(which exists due to the definition of Itō-integrability):
 
 $$\begin{aligned}
     \boxed{
@@ -172,24 +171,16 @@ $$\begin{aligned}
 
 However, $$\mathcal{F}_t$$ says nothing about
 the increment $$(B_{t + h} \!-\! B_t) \sim \mathcal{N}(0, h)$$,
-meaning that the conditional expectation is zero:
+meaning that the conditional expectation is zero for $$t \ge s + h$$:
 
 $$\begin{aligned}
     \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big]
     = 0
-    \qquad \mathrm{for}\; t \ge s + h
 \end{aligned}$$
 
-By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$:
-
-$$\begin{aligned}
-    \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big]
-    = 0
-    \qquad \mathrm{for}\; s \ge t + h
-\end{aligned}$$
-
-This leaves only one case which can be nonzero: $$[t, t\!+\!h] = [s, s\!+\!h]$$.
-Applying the law of total expectation again yields:
+By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$.
+This leaves only one possibly nonzero case: $$[t, t\!+\!h] = [s, s\!+\!h]$$.
+Applying the law of total expectation again:
 
 $$\begin{aligned}
     \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2
@@ -198,15 +189,15 @@ $$\begin{aligned}
     &= \sum_{t = a}^{t = b} \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t^2 (B_{t + h} \!-\! B_t)^2 \Big| \mathcal{F}_t \Big] \bigg]
 \end{aligned}$$
 
-We know $$G_t$$, and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$,
-since the increment is normally distributed, is simply the variance $$h$$:
+We know $$G_t$$,
+and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$ is simply the variance $$h$$:
 
 $$\begin{aligned}
     \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2
     &= \sum_{t = a}^{t = b} \mathbf{E} \big[ G_t^2 \big] h
-    \longrightarrow
-    \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t}
 \end{aligned}$$
+
+Taking the limit $$h \to 0$$ then yields the desired result.
 {% include proof/end.html id="proof-isometry" %}
 
 
@@ -239,7 +230,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We now have everything we need to calculate $$\mathbf{E} [ I_t | \mathcal{F_s} ]$$,
-giving the martingale property:
+leading to the martingale property:
 
 $$\begin{aligned}
     \mathbf{E} \big[ I_t | \mathcal{F}_s \big]
@@ -250,10 +241,10 @@ $$\begin{aligned}
 
 For the existence of $$I_t$$,
 we need $$\mathbf{E}[G_t^2]$$ to be integrable over the target interval,
-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.
+which implies via the Itō isometry that $$\mathbf{E}[I]^2$$ is finite.
+Therefore $$\mathbf{E}[I]$$ is also finite,
+so $$I_t$$ has all the properties of a Martingale
+(since it is trivially $$\mathcal{F}_t$$-adapted).
 {% include proof/end.html id="proof-martingale" %}
 
 
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