commit 7f2667580ba3c6c0036b15fe8a8884759366f58a Author: Bokuan Li Date: Thu Mar 26 15:14:19 2026 -0400 Initial commit. diff --git a/.chktexrc b/.chktexrc new file mode 100644 index 0000000..f0ce6a0 --- /dev/null +++ b/.chktexrc @@ -0,0 +1 @@ +CmdLine { -n24 -n9 -n17 -n25 -n3 } \ No newline at end of file diff --git a/.editorconfig b/.editorconfig new file mode 100644 index 0000000..921dcc9 --- /dev/null +++ b/.editorconfig @@ -0,0 +1,4 @@ +root = true + +[*] +end_of_line = lf \ No newline at end of file diff --git a/.gitea/workflows/compile.yml b/.gitea/workflows/compile.yml new file mode 100644 index 0000000..081902b --- /dev/null +++ b/.gitea/workflows/compile.yml @@ -0,0 +1,18 @@ +name: Compile Project +run-name: Compile the project using spec. +on: [push] +jobs: + Compile: + runs-on: ubuntu-latest + steps: + - name: Check out repository code + uses: actions/checkout@v4 + - name: Add to PATH + run: echo "/root/.nvm/versions/node/v24.14.0/bin" >> $GITHUB_PATH + - name: Copy source + run: | + mkdir -p /srv/builds/${{ github.event.repository.name }} + cp -r . /srv/builds/${{ github.event.repository.name }}/ + - name: Compile Website + run: spec compile --all + working-directory: /srv/builds/${{ github.event.repository.name }} \ No newline at end of file diff --git a/.vscode/project.code-snippets b/.vscode/project.code-snippets new file mode 100644 index 0000000..bcded9b --- /dev/null +++ b/.vscode/project.code-snippets @@ -0,0 +1,164 @@ +{ + "Log to console": { + "scope": "latex", + "prefix": "log", + "body": ["console.info(\"Hello, ${1:World}!\")", "$0"], + "description": "Logs to console" + }, + "Theorem Block": { + "scope": "latex", + "prefix": "thm", + "body": [ + "\\begin{theorem}[$1]", + "\\label{theorem:$2}", + " $3", + "\\end{theorem}", + "$0" + ] + }, + "Definition Block": { + "scope": "latex", + "prefix": "def", + "body": [ + "\\begin{definition}[$1]", + "\\label{definition:$2}", + " $3", + "\\end{definition}", + "$0" + ] + }, + "Lemma Block": { + "scope": "latex", + "prefix": "lem", + "body": [ + "\\begin{lemma}", + "\\label{lemma:$1}", + " $2", + "\\end{lemma}", + "$0" + ] + }, + "Proposition Block": { + "scope": "latex", + "prefix": "prop", + "body": [ + "\\begin{proposition}", + "\\label{proposition:$1}", + " $2", + "\\end{proposition}", + "$0" + ] + }, + "Remark Block": { + "scope": "latex", + "prefix": "rem", + "body": [ + "\\begin{remark}", + "\\label{remark:$1}", + " $2", + "\\end{remark}", + "$0" + ] + }, + "Corollary Block": { + "scope": "latex", + "prefix": "cor", + "body": [ + "\\begin{corollary}", + "\\label{corollary:$1}", + " $2", + "\\end{corollary}", + "$0" + ] + }, + "Proof Block": { + "scope": "latex", + "prefix": "proof", + "body": ["\\begin{proof}", " $1", "\\end{proof}"] + }, + "Bold": { + "scope": "latex", + "prefix": "bf", + "body": ["\\textbf{$1}$0"] + }, + "Italic": { + "scope": "latex", + "prefix": "it", + "body": ["\\textit{$1}$0"] + }, + "Align": { + "scope": "latex", + "prefix": "align", + "body": ["\\begin{align*}", " $1", "\\end{align*}", "$0"] + }, + "Enumerate": { + "scope": "latex", + "prefix": "enum", + "body": ["\\begin{enumerate}", " \\item $1", "\\end{enumerate}", "$0"] + }, + "Section": { + "scope": "latex", + "prefix": "sec", + "body": ["\\section{$1}", "\\label{section:$2}", "", "$0"] + }, + "Subsection": { + "scope": "latex", + "prefix": "subsec", + "body": ["\\subsection{$1}", "\\label{subsection:$2}", "", "$0"] + }, + "Part": { + "scope": "latex", + "prefix": "part", + "body": ["\\part{$1}", "\\label{part:$2}", "", "$0"] + }, + "Chapter": { + "scope": "latex", + "prefix": "chapter", + "body": ["\\chapter{$1}", "\\label{chap:$2}", "", "$0"] + }, + "Citation After Block": { + "scope": "latex", + "prefix": "cite", + "body": ["[{{\\cite[$1]{$2}}}]$0"] + }, + "Mathcal": { + "scope": "latex", + "prefix": "cal", + "body": ["\\mathcal{$1}$0"] + }, + "Mathfrak": { + "scope": "latex", + "prefix": "fk", + "body": ["\\mathfrak{$1}$0"] + }, + "Dual Pairing": { + "scope": "latex", + "prefix": "dp", + "body": ["\\dpb{$1}{$2}$0"] + }, + "Dual Pairing, Default Bracket Size": { + "scope": "latex", + "prefix": "dpn", + "body": ["\\dpn{$1}{$2}$0"] + }, + "d-Dimensional Euclidean Space": { + "scope": "latex", + "prefix": "rd", + "body": ["\\real^d$0"] + }, + "Dyadic Rational Numbers": { + "scope": "latex", + "prefix": "dya", + "body": ["\\mathbb{D}"] + }, + "Rank": { + "scope": "latex", + "prefix": "rank", + "body": ["\\text{rk}"] + }, + "Filtration": { + "scope": "latex", + "prefix": "filt", + "body": ["$\\bracs{\\mathcal{F}_t}$"] + } +} diff --git a/.vscode/settings.json b/.vscode/settings.json new file mode 100644 index 0000000..1d9f0d4 --- /dev/null +++ b/.vscode/settings.json @@ -0,0 +1,11 @@ +{ + "VsCodeTaskButtons.tasks": [ + { + "label": "Watch", + "task": "Watch" + } + ], + "latex.linting.enabled": false, + "latex-workshop.latex.autoBuild.run": "never", + "latex-workshop.latex.texDirs": ["${workspaceFolder}"] +} \ No newline at end of file diff --git a/.vscode/tasks.json b/.vscode/tasks.json new file mode 100644 index 0000000..5736d5f --- /dev/null +++ b/.vscode/tasks.json @@ -0,0 +1,44 @@ +{ + "version": "2.0.0", + "tasks": [ + { + "label": "Build", + "type": "shell", + "command": "npx spec compile", + "windows": { + "options": { + "shell": { + "executable": "cmd.exe", + "args": ["/d", "/c"] + } + } + } + }, + { + "label": "Serve", + "type": "shell", + "command": "npx spec serve", + "windows": { + "options": { + "shell": { + "executable": "cmd.exe", + "args": ["/d", "/c"] + } + } + } + }, + { + "label": "Watch", + "type": "shell", + "command": "npx spec watch", + "windows": { + "options": { + "shell": { + "executable": "cmd.exe", + "args": ["/d", "/c"] + } + } + } + } + ] + } \ No newline at end of file diff --git a/document.tex b/document.tex new file mode 100644 index 0000000..346a48a --- /dev/null +++ b/document.tex @@ -0,0 +1,14 @@ +%\documentclass{report} +\usepackage{amssymb, amsmath, hyperref} +\usepackage{preamble} + +\begin{document} + +\input{./src/index} +%\input{./src/process/index} + +\bibliographystyle{alpha} % We choose the "plain" reference style +\bibliography{refs.bib} % Entries are in the refs.bib file + + +\end{document} diff --git a/preamble.sty b/preamble.sty new file mode 100644 index 0000000..a9281ac --- /dev/null +++ b/preamble.sty @@ -0,0 +1,225 @@ +% \NeedsTeXFormat{LaTeX2e} +% \ProvidesPackage{jerrylicious}[Jerry's Tex Mess] + + +\RequirePackage{amsthm,amssymb,amsfonts,amsmath} +\RequirePackage{enumerate} +\RequirePackage{tikz-cd} + +% ------------- Block Environmets --------------- + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{lemma}[theorem]{Lemma} + +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +% \newtheorem{exercise}[subsection]{Exercise} +% \newtheorem{situation}[subsection]{Situation} + +\newtheorem{rem}[subsection]{Remark} +\newtheorem{remark}[subsection]{Remark} +% \newtheorem{remarks}[subsection]{Remarks} + + +% ------------- References -------------- + +\newcommand{\lemmaautorefname}{Lemma} +\newcommand{\lemautorefname}{Lemma} + + +% ------------------ Shortcuts -------------------------- + +% All kinds of brackets. +\newcommand{\paren}[1]{\left(#1\right)} +\newcommand{\parens}[1]{\left(#1\right)} +\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}} diff --git a/refs.bib b/refs.bib new file mode 100644 index 0000000..4cc872a --- /dev/null +++ b/refs.bib @@ -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} +} diff --git a/spec.db b/spec.db new file mode 100644 index 0000000..7a09b4d Binary files /dev/null and b/spec.db differ diff --git a/spec.toml b/spec.toml new file mode 100644 index 0000000..09470aa --- /dev/null +++ b/spec.toml @@ -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 diff --git a/src/diffusion/brownian.tex b/src/diffusion/brownian.tex new file mode 100644 index 0000000..a0167f4 --- /dev/null +++ b/src/diffusion/brownian.tex @@ -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} + diff --git a/src/diffusion/index.tex b/src/diffusion/index.tex new file mode 100644 index 0000000..5bd370d --- /dev/null +++ b/src/diffusion/index.tex @@ -0,0 +1,5 @@ +\chapter{Stochastic Calculus of Diffusion Processes} +\label{chap:stochastic-calculus} + +\input{./brownian.tex} +\input{./martingale.tex} \ No newline at end of file diff --git a/src/diffusion/martingale.tex b/src/diffusion/martingale.tex new file mode 100644 index 0000000..3ddda8f --- /dev/null +++ b/src/diffusion/martingale.tex @@ -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} + + + diff --git a/src/index.tex b/src/index.tex new file mode 100644 index 0000000..c4c202c --- /dev/null +++ b/src/index.tex @@ -0,0 +1,6 @@ +\part{Diffusion Processes} +\label{part:diffusion} + + + +\input{./diffusion/index.tex} \ No newline at end of file