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-process/index.md | 34 ++++++++++++--------------------
1 file changed, 13 insertions(+), 21 deletions(-)
(limited to 'source/know/concept/ito-process/index.md')
diff --git a/source/know/concept/ito-process/index.md b/source/know/concept/ito-process/index.md
index f192e28..2756e33 100644
--- a/source/know/concept/ito-process/index.md
+++ b/source/know/concept/ito-process/index.md
@@ -61,6 +61,7 @@ since only the current value of $$X_t$$ determines the future,
and $$B_t$$ is also a Markov process.
+
## Itō's lemma
Classically, given $$y \equiv h(x(t), t)$$,
@@ -83,11 +84,8 @@ $$\begin{aligned}
}
\end{aligned}$$
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-
-
-
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+{% include proof/start.html id="proof-lemma" -%}
We start by applying the classical chain rule,
but we go to second order in $$x$$.
This is also valid classically,
@@ -133,14 +131,15 @@ $$\begin{aligned}
Where $$\chi_1^2(\dd{t})$$ is the generalized chi-squared distribution
with one term of variance $$\dd{t}$$.
-
-
+{% include proof/end.html id="proof-lemma" %}
+
The most important application of Itō's lemma
is to perform coordinate transformations,
to make the solution of a given Itō SDE easier.
+
## Coordinate transformations
The simplest coordinate transformation is a scaling of the time axis.
@@ -208,6 +207,7 @@ $$\begin{aligned}
\end{aligned}$$
+
## Existence and uniqueness
It is worth knowing under what condition a solution to a given SDE exists,
@@ -232,11 +232,8 @@ $$\begin{aligned}
}
\end{aligned}$$
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+{% include proof/start.html id="proof-existence" -%}
If we define $$Y_t \equiv X_t^2$$,
then Itō's lemma tells us that the following holds:
@@ -275,9 +272,8 @@ $$\begin{aligned}
\\
&\le (Y_0 + 3 K t) \exp\!\big(3 K t\big)
\end{aligned}$$
+{% include proof/end.html id="proof-existence" %}
-
-
If a solution exists, it is also worth knowing whether it is unique.
Suppose that $$f$$ and $$g$$ satisfy the following inequalities,
@@ -301,11 +297,8 @@ $$\begin{aligned}
}
\end{aligned}$$
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-
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-
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+{% include proof/start.html id="proof-uniqueness" -%}
We define $$D_t \equiv X_t \!-\! Y_t$$ and $$Z_t \equiv D_t^2 \ge 0$$,
together with $$F_t \equiv f(X_t) \!-\! f(Y_t)$$ and $$G_t \equiv g(X_t) \!-\! g(Y_t)$$,
such that Itō's lemma states:
@@ -347,9 +340,8 @@ $$\begin{aligned}
\\
&\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big)
\end{aligned}$$
+{% include proof/end.html id="proof-uniqueness" %}
-
-
Using these properties, it can then be shown
that if all of the above conditions are satisfied,
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