This commit is contained in:
Bokuan Li
4
.editorconfig
Normal file
4
.editorconfig
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@@ -0,0 +1,4 @@
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root = true
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[*]
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end_of_line = lf
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18
.gitea/workflows/compile.yml
Normal file
18
.gitea/workflows/compile.yml
Normal file
@@ -0,0 +1,18 @@
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name: Compile Project
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run-name: Compile the project using spec.
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on: [push]
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jobs:
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Compile:
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runs-on: ubuntu-latest
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steps:
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- name: Check out repository code
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uses: actions/checkout@v4
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- name: Add to PATH
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run: echo "/root/.nvm/versions/node/v24.14.0/bin" >> $GITHUB_PATH
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- name: Copy source
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run: |
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mkdir -p /srv/builds/${{ github.event.repository.name }}
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cp -r . /srv/builds/${{ github.event.repository.name }}/
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- name: Compile Website
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run: spec compile --all
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working-directory: /srv/builds/${{ github.event.repository.name }}
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164
.vscode/project.code-snippets
vendored
Normal file
164
.vscode/project.code-snippets
vendored
Normal file
@@ -0,0 +1,164 @@
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{
|
||||
"Log to console": {
|
||||
"scope": "latex",
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"prefix": "log",
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||||
"body": ["console.info(\"Hello, ${1:World}!\")", "$0"],
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"description": "Logs to console"
|
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},
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"Theorem Block": {
|
||||
"scope": "latex",
|
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"prefix": "thm",
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||||
"body": [
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"\\begin{theorem}[$1]",
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"\\label{theorem:$2}",
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" $3",
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"\\end{theorem}",
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"$0"
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]
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},
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"Definition Block": {
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"scope": "latex",
|
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"prefix": "def",
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"body": [
|
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"\\begin{definition}[$1]",
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"\\label{definition:$2}",
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" $3",
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"\\end{definition}",
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"$0"
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]
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},
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"Lemma Block": {
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"scope": "latex",
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"prefix": "lem",
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"body": [
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"\\begin{lemma}",
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"\\label{lemma:$1}",
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" $2",
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"\\end{lemma}",
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"$0"
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]
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},
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"Proposition Block": {
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||||
"scope": "latex",
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"prefix": "prop",
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"body": [
|
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"\\begin{proposition}",
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"\\label{proposition:$1}",
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" $2",
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"\\end{proposition}",
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"$0"
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||||
]
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},
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"Remark Block": {
|
||||
"scope": "latex",
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"prefix": "rem",
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||||
"body": [
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"\\begin{remark}",
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"\\label{remark:$1}",
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" $2",
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||||
"\\end{remark}",
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"$0"
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||||
]
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},
|
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"Corollary Block": {
|
||||
"scope": "latex",
|
||||
"prefix": "cor",
|
||||
"body": [
|
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"\\begin{corollary}",
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"\\label{corollary:$1}",
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" $2",
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"\\end{corollary}",
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"$0"
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]
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},
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"Proof Block": {
|
||||
"scope": "latex",
|
||||
"prefix": "proof",
|
||||
"body": ["\\begin{proof}", " $1", "\\end{proof}"]
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||||
},
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"Bold": {
|
||||
"scope": "latex",
|
||||
"prefix": "bf",
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||||
"body": ["\\textbf{$1}$0"]
|
||||
},
|
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"Italic": {
|
||||
"scope": "latex",
|
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"prefix": "it",
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"body": ["\\textit{$1}$0"]
|
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},
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"Align": {
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"scope": "latex",
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"prefix": "align",
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"body": ["\\begin{align*}", " $1", "\\end{align*}", "$0"]
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},
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"Enumerate": {
|
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"scope": "latex",
|
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"prefix": "enum",
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||||
"body": ["\\begin{enumerate}", " \\item $1", "\\end{enumerate}", "$0"]
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||||
},
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"Section": {
|
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"scope": "latex",
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"prefix": "sec",
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"body": ["\\section{$1}", "\\label{section:$2}", "", "$0"]
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},
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"Subsection": {
|
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"scope": "latex",
|
||||
"prefix": "subsec",
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||||
"body": ["\\subsection{$1}", "\\label{subsection:$2}", "", "$0"]
|
||||
},
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"Part": {
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"scope": "latex",
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"prefix": "part",
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||||
"body": ["\\part{$1}", "\\label{part:$2}", "", "$0"]
|
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},
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"Chapter": {
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||||
"scope": "latex",
|
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"prefix": "chapter",
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"body": ["\\chapter{$1}", "\\label{chap:$2}", "", "$0"]
|
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},
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"Citation After Block": {
|
||||
"scope": "latex",
|
||||
"prefix": "cite",
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"body": ["[{{\\cite[$1]{$2}}}]$0"]
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},
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"Mathcal": {
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"scope": "latex",
|
||||
"prefix": "cal",
|
||||
"body": ["\\mathcal{$1}$0"]
|
||||
},
|
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"Mathfrak": {
|
||||
"scope": "latex",
|
||||
"prefix": "fk",
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||||
"body": ["\\mathfrak{$1}$0"]
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||||
},
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"Dual Pairing": {
|
||||
"scope": "latex",
|
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"prefix": "dp",
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"body": ["\\dpb{$1}{$2}$0"]
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},
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"Dual Pairing, Default Bracket Size": {
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"scope": "latex",
|
||||
"prefix": "dpn",
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||||
"body": ["\\dpn{$1}{$2}$0"]
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||||
},
|
||||
"d-Dimensional Euclidean Space": {
|
||||
"scope": "latex",
|
||||
"prefix": "rd",
|
||||
"body": ["\\real^d$0"]
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||||
},
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||||
"Dyadic Rational Numbers": {
|
||||
"scope": "latex",
|
||||
"prefix": "dya",
|
||||
"body": ["\\mathbb{D}"]
|
||||
},
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"Rank": {
|
||||
"scope": "latex",
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||||
"prefix": "rank",
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||||
"body": ["\\text{rk}"]
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},
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"Filtration": {
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||||
"scope": "latex",
|
||||
"prefix": "filt",
|
||||
"body": ["$\\bracs{\\mathcal{F}_t}$"]
|
||||
}
|
||||
}
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11
.vscode/settings.json
vendored
Normal file
11
.vscode/settings.json
vendored
Normal file
@@ -0,0 +1,11 @@
|
||||
{
|
||||
"VsCodeTaskButtons.tasks": [
|
||||
{
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"label": "Watch",
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"task": "Watch"
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||||
}
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||||
],
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"latex.linting.enabled": false,
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"latex-workshop.latex.autoBuild.run": "never",
|
||||
"latex-workshop.latex.texDirs": ["${workspaceFolder}"]
|
||||
}
|
||||
44
.vscode/tasks.json
vendored
Normal file
44
.vscode/tasks.json
vendored
Normal file
@@ -0,0 +1,44 @@
|
||||
{
|
||||
"version": "2.0.0",
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||||
"tasks": [
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{
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"label": "Build",
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"type": "shell",
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"command": "npx spec compile",
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"windows": {
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"options": {
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"shell": {
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||||
"executable": "cmd.exe",
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||||
"args": ["/d", "/c"]
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||||
}
|
||||
}
|
||||
}
|
||||
},
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||||
{
|
||||
"label": "Serve",
|
||||
"type": "shell",
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||||
"command": "npx spec serve",
|
||||
"windows": {
|
||||
"options": {
|
||||
"shell": {
|
||||
"executable": "cmd.exe",
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||||
"args": ["/d", "/c"]
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||||
}
|
||||
}
|
||||
}
|
||||
},
|
||||
{
|
||||
"label": "Watch",
|
||||
"type": "shell",
|
||||
"command": "npx spec watch",
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||||
"windows": {
|
||||
"options": {
|
||||
"shell": {
|
||||
"executable": "cmd.exe",
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||||
"args": ["/d", "/c"]
|
||||
}
|
||||
}
|
||||
}
|
||||
}
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||||
]
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||||
}
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14
document.tex
Normal file
14
document.tex
Normal file
@@ -0,0 +1,14 @@
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%\documentclass{report}
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||||
\usepackage{amssymb, amsmath, hyperref}
|
||||
\usepackage{preamble}
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||||
|
||||
\begin{document}
|
||||
|
||||
\input{./src/index}
|
||||
%\input{./src/process/index}
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|
||||
\bibliographystyle{alpha} % We choose the "plain" reference style
|
||||
\bibliography{refs.bib} % Entries are in the refs.bib file
|
||||
|
||||
|
||||
\end{document}
|
||||
225
preamble.sty
Normal file
225
preamble.sty
Normal file
@@ -0,0 +1,225 @@
|
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% \NeedsTeXFormat{LaTeX2e}
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% \ProvidesPackage{jerrylicious}[Jerry's Tex Mess]
|
||||
|
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|
||||
\RequirePackage{amsthm,amssymb,amsfonts,amsmath}
|
||||
\RequirePackage{enumerate}
|
||||
\RequirePackage{tikz-cd}
|
||||
|
||||
% ------------- Block Environmets ---------------
|
||||
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||||
\newtheorem{theorem}{Theorem}[section]
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||||
\newtheorem{proposition}[theorem]{Proposition}
|
||||
\newtheorem{corollary}[theorem]{Corollary}
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
|
||||
\newtheorem{definition}[theorem]{Definition}
|
||||
\newtheorem{example}[theorem]{Example}
|
||||
% \newtheorem{exercise}[subsection]{Exercise}
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||||
% \newtheorem{situation}[subsection]{Situation}
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||||
|
||||
\newtheorem{rem}[subsection]{Remark}
|
||||
\newtheorem{remark}[subsection]{Remark}
|
||||
% \newtheorem{remarks}[subsection]{Remarks}
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||||
|
||||
|
||||
% ------------- References --------------
|
||||
|
||||
\newcommand{\lemmaautorefname}{Lemma}
|
||||
\newcommand{\lemautorefname}{Lemma}
|
||||
|
||||
|
||||
% ------------------ Shortcuts --------------------------
|
||||
|
||||
% All kinds of brackets.
|
||||
\newcommand{\paren}[1]{\left(#1\right)}
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||||
\newcommand{\parens}[1]{\left(#1\right)}
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\newcommand{\bracs}[1]{\left\{#1\right\}}
|
||||
\newcommand{\braks}[1]{\left[#1\right]}
|
||||
\newcommand{\angles}[1]{\left\langle#1\right\rangle}
|
||||
\newcommand{\abs}[1]{\left|#1\right|}
|
||||
\newcommand{\bracsn}[1]{\{{#1}\}}
|
||||
\newcommand{\anglesn}[1]{\langle {#1} \rangle}
|
||||
|
||||
% Probability
|
||||
\newcommand{\ev}{\mathbb{E}}
|
||||
\newcommand{\var}[1]{\text{Var}\paren{#1}}
|
||||
\newcommand{\cov}[1]{\text{Cov}\paren{#1}}
|
||||
\newcommand{\sig}[1]{\sigma_{#1}^2}
|
||||
\newcommand{\bp}{\mathbf{P}}
|
||||
|
||||
% Complex numbers
|
||||
\newcommand{\re}[1]{\text{Re}\paren{#1}}
|
||||
\newcommand{\im}[1]{\text{Im}\paren{#1}}
|
||||
\newcommand{\sgn}{\text{sgn}}
|
||||
|
||||
% Bold stuff for algebra.
|
||||
\newcommand{\zero}{\mathbf{0}}
|
||||
\newcommand{\one}{\mathbf{1}}
|
||||
|
||||
% Number Systems/Algebraic Structures
|
||||
\newcommand{\polyfield}[1]{\mathbb{F}[#1]}
|
||||
\newcommand{\field}{\mathbb{F}}
|
||||
\newcommand{\nat}{\mathbb{N}}
|
||||
\newcommand{\natp}{\mathbb{N}^+}
|
||||
\newcommand{\natz}{\mathbb{N}_0}
|
||||
\newcommand{\integer}{\mathbb{Z}}
|
||||
\newcommand{\complex}{\mathbb{C}}
|
||||
\newcommand{\real}{\mathbb{R}}
|
||||
\newcommand{\rd}{\real^d}
|
||||
\newcommand{\realp}{\mathbb{R}_{>0}} % Real positive
|
||||
\newcommand{\realnn}{\mathbb{R}_{\ge 0}} % Real non-negative
|
||||
\newcommand{\rational}{\mathbb{Q}}
|
||||
\newcommand{\quot}[1]{\integer/{#1}\integer}
|
||||
\newcommand{\polyfields}[1]{F[{#1}_1, \cdots, {#1}_n]}
|
||||
\newcommand{\polyfieldf}[1]{F({#1}_1, \cdots, {#1}_n)}
|
||||
\newcommand{\gal}[1]{\text{Gal}\paren{#1}}
|
||||
\newcommand{\aut}[1]{\text{Aut}\paren{#1}}
|
||||
\newcommand{\fa}{\mathfrak{a}}
|
||||
\newcommand{\fb}{\mathfrak{b}}
|
||||
\newcommand{\fv}{\mathfrak{v}}
|
||||
|
||||
% Sequences and Limits
|
||||
\newcommand{\limv}[1]{\lim_{#1 \to \infty}}
|
||||
\newcommand{\seq}[1]{\bracs{#1}_{1}^{\infty}}
|
||||
\newcommand{\limsupd}[1]{\underset{#1}{\ol{\lim}}}
|
||||
\newcommand{\liminfd}[1]{\underset{#1}{\underline{\lim}}}
|
||||
\newcommand{\seqi}[1]{\bracs{{#1}_{i}}_{i \in I}}
|
||||
\newcommand{\seqj}[1]{\bracs{{#1}_{j}}_{j \in J}}
|
||||
\newcommand{\drarrow}{\searrow}
|
||||
\newcommand{\downto}{\searrow}
|
||||
\newcommand{\upto}{\nearrow}
|
||||
\newcommand{\eps}{\varepsilon}
|
||||
% Optional upper index argument.
|
||||
\newcommand{\seqf}[2][n]{\bracs{#2}_{1}^{#1}}
|
||||
\newcommand{\seqfz}[2][n]{\bracs{#2}_{0}^{#1}}
|
||||
|
||||
% Isomorphisms
|
||||
\newcommand{\isoto}{\tilde{\rightarrow}}
|
||||
\newcommand{\iso}{\cong}
|
||||
|
||||
\newcommand{\modulo}{\text{mod }}
|
||||
|
||||
% Algebra properties
|
||||
\newcommand{\ord}{\text{Ord }}
|
||||
\newcommand{\orb}[1]{\text{Orb}\paren{#1}}
|
||||
\newcommand{\stab}[1]{\text{Stab}\paren{#1}}
|
||||
\newcommand{\rank}{\text{rank }}
|
||||
\newcommand{\inp}{\angles{\cdot, \cdot}}
|
||||
\newcommand{\eig}{\text{Eig}}
|
||||
\newcommand{\proj}{\text{proj}}
|
||||
\newcommand{\tr}{\text{tr}}
|
||||
\newcommand{\spec}[1]{\text{Spec}\paren{#1}}
|
||||
\newcommand{\ba}{\mathbf{A}}
|
||||
|
||||
% Dual Pairings & Inner Products
|
||||
|
||||
\newcommand{\dpb}[2]{\angles{#1}_{#2}} % Dual Pairing with a specific space. B because it's apparently already defined in tex.
|
||||
\newcommand{\dpn}[2]{\langle {#1} \rangle_{#2}} % Dual Pairing, without scaling the braces.
|
||||
\newcommand{\dprd}[1]{\dpb{#1}{\rd}} % R^d dual pairing.
|
||||
\newcommand{\dprdn}[1]{\dpn{#1}{\rd}} % R^d dual pairing.
|
||||
\newcommand{\dpd}[1]{\dpb{#1}{\mathcal{D}}} % Distribution dual pairing
|
||||
\newcommand{\dpdn}[1]{\dpn{#1}{\mathcal{D}}}
|
||||
\newcommand{\dpe}[1]{\dpb{#1}{E}} % E for Banach.
|
||||
\newcommand{\dpen}[1]{\dpn{#1}{E}} % E for Banach.
|
||||
\newcommand{\dph}[1]{\dpb{#1}{H}} % H for Hilbert.
|
||||
\newcommand{\dphn}[1]{\dpn{#1}{H}} % H for Hilbert.
|
||||
|
||||
% Floor and ceiling
|
||||
\newcommand{\fl}[1]{\left\lfloor #1 \right\rfloor}
|
||||
\newcommand{\cl}[1]{\left\lceil #1 \right\rceil}
|
||||
|
||||
% Topology
|
||||
\newcommand{\topo}{\mathcal{T}}
|
||||
\newcommand{\inte}{\text{int}}
|
||||
\newcommand{\exte}{\text{ext}}
|
||||
\newcommand{\diam}{\text{diam}}
|
||||
\newcommand{\net}[1]{\angles{{#1}_{\alpha}}_{\alpha \in A}}
|
||||
\newcommand{\netb}[1]{\angles{{#1}_{\beta}}_{\beta \in B}}
|
||||
\newcommand{\supp}[1]{\text{supp}\paren{#1}}
|
||||
\newcommand{\fF}{\mathfrak{F}}
|
||||
\newcommand{\fS}{\mathfrak{S}}
|
||||
\newcommand{\fB}{\mathfrak{B}}
|
||||
\newcommand{\fU}{\mathfrak{U}}
|
||||
\newcommand{\fV}{\mathfrak{V}}
|
||||
\newcommand{\fG}{\mathfrak{G}}
|
||||
\newcommand{\bs}{\mathbf{S}}
|
||||
|
||||
% Curly Letters
|
||||
\newcommand{\alg}{\mathcal{A}}
|
||||
\newcommand{\agb}[1]{\mathcal{M}\paren{#1}}
|
||||
\newcommand{\cm}{\mathcal{M}}
|
||||
\newcommand{\cf}{\mathcal{F}}
|
||||
\newcommand{\ce}{\mathcal{E}}
|
||||
\newcommand{\ci}{\mathcal{I}}
|
||||
\newcommand{\pow}[1]{\mathcal{P}\paren{#1}}
|
||||
\newcommand{\cb}{\mathcal{B}}
|
||||
\newcommand{\cn}{\mathcal{N}}
|
||||
\newcommand{\lms}{\mathcal{L}}
|
||||
\newcommand{\ch}{\mathcal{H}}
|
||||
\newcommand{\cg}{\mathcal{H}}
|
||||
\renewcommand{\cl}{\mathcal{L}}
|
||||
\newcommand{\cc}{\mathcal{C}}
|
||||
\newcommand{\ct}{\mathcal{T}}
|
||||
\newcommand{\cx}{\mathcal{X}}
|
||||
\newcommand{\cy}{\mathcal{Y}}
|
||||
\newcommand{\cs}{\mathcal{S}}
|
||||
\newcommand{\cd}{\mathcal{D}}
|
||||
\newcommand{\calr}{\mathcal{R}}
|
||||
\newcommand{\scp}{\mathscr{P}}
|
||||
|
||||
% Jokes
|
||||
\newcommand{\lol}{\boxed{\text{LOL.}}}
|
||||
\newcommand{\ez}{\boxed{\mathbb{EZ}}}
|
||||
|
||||
% Colours
|
||||
\newcommand{\pblue}[1]{\textcolor[rgb]{0, 0.44, 0.75}{#1}}
|
||||
\newcommand{\poran}[1]{\textcolor{orange}{#1}}
|
||||
\newcommand{\pb}[1]{\pblue{#1}}
|
||||
\newcommand{\po}[1]{\poran{#1}}
|
||||
\newcommand{\pg}[1]{\textcolor[rgb]{0, 0.69, 0.31}{#1}}
|
||||
\newcommand{\pp}[1]{\textcolor[rgb]{0.35, 0.25, 0.64}{#1}}
|
||||
\newcommand{\ps}[1]{\textcolor[rgb]{0.58, 0.25, 0.45}{#1}}
|
||||
|
||||
% Formatting
|
||||
\newcommand{\ol}[1]{\overline{#1}}
|
||||
\newcommand{\ul}[1]{\underline{#1}}
|
||||
\newcommand{\wh}[1]{\widehat{#1}}
|
||||
\newcommand{\td}[1]{\widetilde{#1}}
|
||||
|
||||
\newcommand{\norm}[1]{\abs{\abs{#1}}}
|
||||
\newcommand{\norms}[2]{\abs{\abs{#1}}_{#2}}
|
||||
\newcommand{\normn}[1]{||{#1}||}
|
||||
\newcommand{\normns}[2]{||{#1}||_{#1}}
|
||||
\newcommand{\normrd}[1]{\norms{#1}{\real^d}}
|
||||
\newcommand{\normrdn}[1]{\normns{#1}{\real^d}}
|
||||
|
||||
% Category Shenanigans
|
||||
\newcommand{\mf}[1]{\mathfrak{#1}}
|
||||
\newcommand{\catc}{\mathfrak{C}}
|
||||
\newcommand{\cata}{\mathfrak{A}}
|
||||
\newcommand{\obj}[1]{\text{Ob}\paren{#1}}
|
||||
\newcommand{\mor}[1]{\text{Mor}\paren{#1}}
|
||||
|
||||
% Tangent Space Shenanigans
|
||||
\newcommand{\ppi}{\frac{\partial}{\partial x^i}}
|
||||
\newcommand{\ppj}{\frac{\partial}{\partial y^j}}
|
||||
\newcommand{\ppip}{\ppi\bigg\vert_p}
|
||||
\newcommand{\vf}{\mathfrak{X}}
|
||||
\newcommand{\grad}[1]{\text{grad}\paren{#1}}
|
||||
\newcommand{\Grad}[1]{\text{Grad}\paren{#1}}
|
||||
\renewcommand{\div}[1]{\text{div}\paren{#1}}
|
||||
|
||||
% Random spaces
|
||||
\newcommand{\laut}[1]{\text{Laut}({#1})}
|
||||
\newcommand{\loci}{L^1_{\text{loc}}}
|
||||
\newcommand{\symbi}{L^2_{\text{sym}}}
|
||||
\newcommand{\met}{\text{Met}}
|
||||
\newcommand{\ri}{\text{Ri}}
|
||||
\newcommand{\ind}[1]{\text{ind}\paren{#1}}
|
||||
\newcommand{\scl}{\mathscr{L}}
|
||||
\newcommand{\sch}{\mathscr{H}}
|
||||
\newcommand{\frl}{\mathfrak{L}}
|
||||
|
||||
% Real or Complex Numbers
|
||||
\newcommand{\RC}{\bracs{\real, \complex}}
|
||||
10
refs.bib
Normal file
10
refs.bib
Normal file
@@ -0,0 +1,10 @@
|
||||
@book{Diffusion,
|
||||
title={Multidimensional Diffusion Processes},
|
||||
author={Stroock, D.W. and Varadhan, S.R.S.},
|
||||
isbn={9783540903536},
|
||||
lccn={96027093},
|
||||
series={Grundlehren der mathematischen Wissenschaften},
|
||||
url={https://books.google.ca/books?id=DuDsmoyqCy4C},
|
||||
year={1997},
|
||||
publisher={Springer Berlin Heidelberg}
|
||||
}
|
||||
21
spec.toml
Normal file
21
spec.toml
Normal file
@@ -0,0 +1,21 @@
|
||||
database = "spec.db"
|
||||
document = "document.tex"
|
||||
siteTitle = "Unnamed Website"
|
||||
|
||||
[compiler]
|
||||
compileAll = false
|
||||
redoTags = false
|
||||
indirectReferences = true
|
||||
|
||||
[website]
|
||||
font = "roboto"
|
||||
fontSize = 16
|
||||
lineHeight = 1.3
|
||||
textAlign = "left"
|
||||
primaryColour = "blue"
|
||||
neutralColour = "grey"
|
||||
searchLimit = 16
|
||||
maxSearchPages = 48
|
||||
recentChanges = 10
|
||||
tableOfContentsDepth = 2
|
||||
advertiseSpec = true
|
||||
71
src/diffusion/brownian.tex
Normal file
71
src/diffusion/brownian.tex
Normal file
@@ -0,0 +1,71 @@
|
||||
\section{Brownian Motion}
|
||||
\label{section:brownian-motion}
|
||||
|
||||
\begin{theorem}
|
||||
\label{theorem:brownian-martingales}
|
||||
Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $\bracs{B_t|t \ge 0}$ be a $\real^d$-valued, $\bracs{\cf_t}$-progressively measurable process such that $B_0 = 0$ almost surely, then the following are equivalent:
|
||||
\begin{enumerate}
|
||||
\item $\bracs{B_t|t \ge 0}$ is a Brownian motion.
|
||||
\item For each $\xi \in \real^d$, the process
|
||||
\[
|
||||
X^\theta_t = \exp\braks{i \dpn{B_t, \xi}{\real^d} + \frac{1}{2}\norm{\xi}_{\real^d}^2t}
|
||||
\]
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
\item For each $\phi \in BC^2(\real^d)$, the process
|
||||
\[
|
||||
Y^\phi_t = \phi(B_t) - \frac{1}{2}\int_0^t \Delta f(B_s)ds
|
||||
\]
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
\begin{proof}[Proof {{\cite[Theorem 4.1.1]{Diffusion}}}. ]
|
||||
(1) $\Rightarrow$ (2): For each $1 \le s \le t$ and $\xi \in \real^d$,
|
||||
\begin{align*}
|
||||
\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}} \big| \cf_s} &= \ev\braks{e^{i \dpn{B_s, \xi}{\real^d}}e^{i \dpn{B_t - B_s, \xi}{\real^d}} \big| \cf_s} \\
|
||||
&= e^{i \dpn{B_s, \xi}{\real^d}}e^{-(t - s)\norm{\xi}_{\real^d}^2/2}
|
||||
\end{align*}
|
||||
|
||||
Therefore
|
||||
\[
|
||||
\ev\braks{e^{i \dpn{B_t, \xi}{\real^d} + t\norm{\xi}^2/2} \big| \cf_s} = e^{i \dpn{B_s, \xi}{\real^d} + s\norm{\xi}_{\real^d}^2/2}
|
||||
\]
|
||||
|
||||
(2) $\Rightarrow$ (3): For each $0 \le s < t$ and $\xi \in \real^d$,
|
||||
\begin{align*}
|
||||
\frac{d}{dt}\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}} \big| \cf_s} &= -\frac{\norm{\xi}_{\real^d}^2}{2} e^{i \dpn{B_s, \xi}{\real^d}}e^{-(t - s)\norm{\xi}_{\real^d}^2/2}
|
||||
\end{align*}
|
||||
|
||||
Therefore for any $0 \le s < t$,
|
||||
\begin{align*}
|
||||
\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}}-e^{i \dpn{B_s, \xi}{\real^d}} \big| \cf_s} &= -\frac{\norm{\xi}_{\real^d}^2}{2}\int_s^t e^{i \dpn{B_s, \xi}{\real^d}}e^{-(r - s)\norm{\xi}_{\real^d}^2/2}dr \\
|
||||
&= -\frac{\norm{\xi}_{\real^d}^2}{2}\int_s^t \ev\braks{e^{i \dpn{B_r, \xi}{\real^d}} \big | \cf_{s}}dr
|
||||
\end{align*}
|
||||
|
||||
If $\phi_\xi(x) = e^{-i \dpn{x, \xi}{\real^d}}$, then
|
||||
\[
|
||||
\ev\braks{e^{i \dpn{B_t, \xi}{\real^d}}-e^{i \dpn{B_s, \xi}{\real^d}} \big| \cf_s} = \ev\braks{\int_s^t \frac{1}{2}(\Delta \phi_\xi)(B_r)dd \bigg | \cf_s}
|
||||
\]
|
||||
|
||||
so the process
|
||||
\[
|
||||
Z^\xi_t = e^{i \dpn{B_t, \xi}{\real^d}} + \int_0^t \frac{1}{2}(\Delta \phi_\xi)(B_r)dr
|
||||
\]
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
|
||||
Let $\phi \in \mathcal{S}(\real^d)$, then by the Fourier Inversion Theorem, there exists $\psi \in \mathcal{S}(\real^d)$ such that for any $x \in \real^d$,
|
||||
\[
|
||||
\phi(x) = \int e^{i \dpn{\xi, x}{\real^d}}\psi(\xi)d\xi
|
||||
\]
|
||||
|
||||
In which case, for any $t \ge 0$,
|
||||
\[
|
||||
\phi(B_t) + \int_0^t \frac{1}{2}(\Delta \phi)(B_r)dr
|
||||
= \int_{\real^d} e^{i \dpn{\xi, B_t}{\real^d}}\psi(\xi)d\xi + \int_{\real^d}\int_0^t \frac{1}{2}(\Delta \phi_\xi)(B_r) \psi(\xi) dr d\xi
|
||||
\]
|
||||
|
||||
By the Dominated Convergence Theorem, the above also holds for any $\phi \in BC^2(\real^d)$. Therefore the process $\bracsn{Y^\phi_t|t \ge 0}$ is a $\bracs{\mathcal{F}_t}$-martingale as well.
|
||||
\end{proof}
|
||||
|
||||
5
src/diffusion/index.tex
Normal file
5
src/diffusion/index.tex
Normal file
@@ -0,0 +1,5 @@
|
||||
\chapter{Stochastic Calculus of Diffusion Processes}
|
||||
\label{chap:stochastic-calculus}
|
||||
|
||||
\input{./brownian.tex}
|
||||
\input{./martingale.tex}
|
||||
194
src/diffusion/martingale.tex
Normal file
194
src/diffusion/martingale.tex
Normal file
@@ -0,0 +1,194 @@
|
||||
\section{Equivalence of Certain Martingales}
|
||||
\label{section:martingale-equivalence}
|
||||
|
||||
|
||||
% I made this one up.
|
||||
|
||||
\begin{definition}[Random Differential Operator]
|
||||
\label{definition:random-differential-operator}
|
||||
Let $(\Omega, \cf, \bp)$ be a filtered probability space, $s \ge 0$, and $k \in \natp$. For each multi-index $\alpha \in \nat_0^d$ with $\abs{\alpha} \le k$, let $a_\alpha: [s, \infty) \times \Omega \to \real$ be measurable function. For each $f \in C^k(\real^d)$ and $t \ge s$, let
|
||||
\[
|
||||
[L_t(\omega)f](x) = \sum_{\abs{\alpha} \le k} a_\alpha(t, \omega) \partial^\alpha f(x)
|
||||
\]
|
||||
|
||||
then $L = \sum_{|\alpha| \le k}a_\alpha \partial^\alpha$ is a \textbf{random differential operator} of order $k$ with coefficients $\bracs{a_\alpha}_{|\alpha| \le k}$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Progressively Measurable Random Differential Operator]
|
||||
\label{definition:random-do-progressively-measurable}
|
||||
Let $(\Omega, \cf, \bp)$ be a filtered probability space and $L = \sum_{|\alpha| \le k}a_\alpha$ be a random differential operator on $\Omega$, then $L$ is \textbf{progressively measurable} with respect to $\bracs{\mathcal{F}_t}$ if for each $\alpha \in \nat_0$ with $|\alpha| \le k$, $a_\alpha: [s, \infty) \times \Omega \to \real$ is progressively measurable with respect to $\bracs{\mathcal{F}_t}$.
|
||||
\end{definition}
|
||||
|
||||
|
||||
\begin{lemma}
|
||||
\label{lemma:random-diff-process}
|
||||
Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $s \ge 0$, $k \in \natp$, $L = \sum_{|\alpha| \le k}a_\alpha \partial^\alpha$ be a $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, $f \in C^k(\real^d)$, and $\bracs{X_t|t \ge 0}$ be a $\real^d$-valued progressively measurable process, then
|
||||
\[
|
||||
Y_t(\omega) = (L_t(\omega)f)(X_t(\omega))
|
||||
\]
|
||||
|
||||
is a progressively measurable process.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[Integration by Parts]
|
||||
\label{theorem:martingale-ibp}
|
||||
Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $\bracs{X_t}$ be a $\bracs{\mathcal{F}_t}$-martingale, and $\phi: [0, \infty) \times \Omega \to \complex$ be a continuous, progressively measurable function. If:
|
||||
\begin{enumerate}
|
||||
\item For every $\omega \in \Omega$, $\phi(\cdot, \omega) \in BV_{\text{loc}}([0, \infty))$.
|
||||
\item For all $t \ge 0$,
|
||||
\[
|
||||
\ev\braks{\sup_{0 \le s \le t}|X_s|(|\phi|(s) + |\phi|(t))} < \infty
|
||||
\]
|
||||
\end{enumerate}
|
||||
|
||||
then the process
|
||||
\[
|
||||
Y_t = X_t \phi(t) - \int_0^t X_r \phi(dr)
|
||||
\]
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
\end{theorem}
|
||||
|
||||
|
||||
\begin{theorem}[Equivalence of Martingales]
|
||||
\label{theorem:equivalence-of-martingales}
|
||||
Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space,
|
||||
\[
|
||||
L_t(\omega)u = \frac{1}{2}\dpn{A_t(\omega), u}{\real^{d \times d}} + \dpn{B_t(\omega), u}{\real^d}
|
||||
\]
|
||||
|
||||
be a second-order $\bracs{\mathcal{F}_t}$-progressively measurable random differential operator, and $\bracs{X_t|t \ge 0}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable process with continuous sample paths, then the following are equivalent:
|
||||
\begin{enumerate}
|
||||
\item For every $f \in C_c^\infty(\real^d)$, the process
|
||||
\[
|
||||
E_t = f(X_t) - \int_0^t (L_rf)(X_r)dr
|
||||
\]
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
\item For every $f \in C_b^{1, 2}([0, \infty) \times \real^d)$,
|
||||
\[
|
||||
H_t = f(t, X_t) - \int_0^t [(\partial_r + L_r)f](r, X_r)dr
|
||||
\]
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
\item For each uniformly positive $f \in C_b^{1, 2}([0, \infty) \times \real^d)$,
|
||||
\[
|
||||
X_t = f(t, X_t)\exp\braks{-\int_0^t \frac{(\partial_r + L_r)f}{f}(r, X_r)dr}
|
||||
\]
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
\item For each $x \in \real^d$ and $g \in C_b^{1, 2}([0, \infty) \times \real^d)$, the process
|
||||
\begin{align*}
|
||||
Y_t^{x, g} &= \exp\braks{\dpn{x, X_t - X_0 - \int_0^t b(r)dr}{\real^d} + g(t, X_t)} \\
|
||||
&\cdot \exp\braks{-\frac{1}{2}\int_0^t \dpn{x + Dg, A(r)(x + Dg)}{\real^d}(r, X_r)dr} \\
|
||||
&\cdot \exp\braks{\int_0^t [(\partial_r + L_r)g](r, X_r)}dr
|
||||
\end{align*}
|
||||
|
||||
is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
\item For each $\theta \in \real^d$,
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
\end{theorem}
|
||||
\begin{proof}[Proof {{\cite[Theorem 4.2.1]{Diffusion}}}. ]
|
||||
(1) $\Rightarrow$ (2): Let $0 \le s \le t$ and $f \in C_c^\infty([0, \infty) \times \real^d)$, then by the Fundamental Theorem of Calculus,
|
||||
\begin{align*}
|
||||
f(t, X_t) - f(s, X_s) &= f(t, X_t) - f(s, X_t) + f(s, X_t) - f(s, X_s) \\
|
||||
&= \int_s^t \partial_rf(r, X_t)dr + f(s, X_t) - f(s, X_s)
|
||||
\end{align*}
|
||||
|
||||
where by (1),
|
||||
\[
|
||||
\ev\braks{f(s, X_t) - f(s, X_s) |\cf_s} = \ev\braks{\int_s^t (L_rf)(s, X_r)dr \bigg | \cf_s}
|
||||
\]
|
||||
|
||||
By the Fundamental Theorem of Calculus,
|
||||
\begin{align*}
|
||||
\int_s^t (\partial_rf)(r, X_t)dr &= \int_s^t (\partial_r f)(r, X_r)dr
|
||||
+ \int_s^t (\partial_r f)(r, X_t) - (\partial_r f)(r, X_r)dr \\
|
||||
&= \int_s^t (\partial_r f)(r, X_r)dr
|
||||
+ \int_s^t (\partial_r f)(r, X_t) - (\partial_r f)(r, X_r)dr \\
|
||||
\end{align*}
|
||||
|
||||
By (1) applied to $(\partial_r f)$,
|
||||
\[
|
||||
\ev\braks{\int_s^t (\partial_rf)(r, X_t)dr \bigg | \cf_s} =
|
||||
\braks{\int_s^t (\partial_r f)(r, X_r)dr
|
||||
+ \int_s^t \int_r^t (\partial_r f)(r, X_u)du dr \bigg | \cf_s}
|
||||
\]
|
||||
|
||||
Similarly, by the Fundamental Theorem of Calculus,
|
||||
\begin{align*}
|
||||
\int_s^t (L_rf)(s, X_r)dr &= \int_s^t (L_rf)(r, X_r)dr
|
||||
+ \int_s^t (L_rf)(s, X_r) - (L_rf)(r, X_r)dr \\
|
||||
&= \int_s^t(L_rf)(r, X_r)dr - \int_s^t \int_s^r (\partial_rL_rf)(u, X_r)dudr
|
||||
\end{align*}
|
||||
|
||||
Since the two iterated integrals are over the same region,
|
||||
\[
|
||||
\ev\braks{f(t, X_t) - f(s, X_s) | \cf_s} = \ev\braks{\int_s^t (\partial_r + L_rf)(r, X_r)dr \bigg | \cf_s}
|
||||
\]
|
||||
|
||||
Therefore $\bracs{E_t}$ is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
|
||||
(2) $\Rightarrow$ (3): Let $f \in C_b^{1, 2}([0, \infty) \times \real^d)$ be uniformly positive. Define
|
||||
\[
|
||||
Z_t = f(t, X_t) - \int_0^t (\partial_rf + L_rf)(r, X_r)dr
|
||||
\]
|
||||
|
||||
and
|
||||
\[
|
||||
\phi_t = \exp\braks{-\int_0^t \frac{\partial_rf + L_rf}{f}(r, X_r)dr}
|
||||
\]
|
||||
|
||||
then using the by parts formula for Riemann-Stieltjes integrals,
|
||||
\begin{align*}
|
||||
Z_t\phi_t - \int_0^t Z_s \phi(ds) &= Z_0\phi_0 + \int_0^t \phi_s Z(ds) \\
|
||||
&= Z_0\phi_0 + \int_0^t \phi_s f(ds, X_{ds}) + \int_0^t \phi_s \cdot (\partial_s f L_sf)(s, X_s)ds
|
||||
\end{align*}
|
||||
|
||||
Since
|
||||
\[
|
||||
\partial_s \phi_s = -\phi_s \frac{\partial_s f + L_sf}{f}(s, X_s)
|
||||
\]
|
||||
|
||||
Using the by parts formula again,
|
||||
|
||||
\begin{align*}
|
||||
Z_t\phi_t - \int_0^t Z_s \phi(ds) &= Z_0\phi_0 + \int_0^t \phi_s f(ds, X_{ds}) + \int_0^t f(s, X_s)\phi(ds) \\
|
||||
&= Z_0\phi_0 + f(t, X_t)\phi_t - f(0, X_0)\phi_0 = f(t, X_t)\phi_t \\
|
||||
&= f(t, X_t)\exp\braks{-\int_0^t \frac{\partial_rf + L_rf}{f}(r, X_r)dr}
|
||||
\end{align*}
|
||||
|
||||
Therefore \autoref{theorem:martingale-ibp} implies that the above process is a $\bracs{\mathcal{F}_t}$-martingale.
|
||||
|
||||
(3) $\Rightarrow$ (4): Let $x \in \real^d$ and $g \in C_b^{1, 2}([0, \infty) \times \real^d)$. Define
|
||||
\[
|
||||
f: [0, \infty) \times \real^d \to \real \quad (t, y) \mapsto \exp(\dpn{x, y}{\real^d} + g(t, x))
|
||||
\]
|
||||
|
||||
then (3) applied to $f$ yields that $\bracs{Y_t|t \ge 0}$ is a $\bracs{\mathcal{F}_t}$-martingale. However, since $f$ is unbounded and not uniformly positive, a direct application is ineffective.
|
||||
|
||||
Let $\seq{f_n} \subset C_b^{1, 2}([0, \infty) \times \real^d)$ be uniformly positive such that $f_n|_{\bracs{|y| \le n}} = f$. Define
|
||||
\[
|
||||
\tau_n = \inf\bracs{t \ge 0 \bigg | \sup_{0 \le s \le t}|X_t(u)| \ge n} \wedge n
|
||||
\]
|
||||
|
||||
then $\bracsn{Y_{t \wedge \tau_n}^{x, g}}$ is a $\bracs{\mathcal{F}_t}$-martingale with $Y_{t \wedge \tau_n}^{x, g} \to Y_{t}^{x, g}$ almost surely as $n \to \infty$. In addition,
|
||||
\[
|
||||
(Y_{t \wedge \tau_n}^{x, g})^2 \le Y_{t \wedge \tau_n}^{2x, 2g} \cdot \exp\braks{\int_0^{t \wedge \tau_n} \dpn{x + Dg, A_s(x + Dg) }{\real^d}(s, X_s)ds} \le CY_{t \wedge \tau_n}^{2x, 2g}
|
||||
\]
|
||||
|
||||
where $\ev\braks{Y_{t \wedge \tau_n}^{2x, 2g}} = f(s, X_s)^2 \le \exp(2\norm{g}_u)$. Therefore $\bracsn{Y_{t \wedge \tau_n}^{x, g}}$ is bounded in $L^2$, uniformly integrable in $L^1$, and converges to $Y_{t}^{x, g}$ in $L^1$.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
6
src/index.tex
Normal file
6
src/index.tex
Normal file
@@ -0,0 +1,6 @@
|
||||
\part{Diffusion Processes}
|
||||
\label{part:diffusion}
|
||||
|
||||
|
||||
|
||||
\input{./diffusion/index.tex}
|
||||
Reference in New Issue
Block a user