Housekeeping.

spec.toml

@@ -1,6 +1,6 @@
 database = "spec.db"
 document = "document.tex"
-siteTitle = "Unnamed Website"
+siteTitle = "Brownian Motion and Stochastic Calculus"
 
 [compiler]
 compileAll = false

src/calculus/index.tex

@@ -1,4 +1,4 @@
-\chapter{Integral Manifolds/Shenanigans}
+\chapter{Integral Manifolds}
 \label{chap:integral}
 
 \input{./integrable.tex}

src/index.tex

@@ -1,9 +1,14 @@
-\part{Diffusion Processes}
+\part{Stochastic Processes}
 \label{part:diffusion}
 
 \input{./aws/index.tex}
 \input{./operator/index.tex}
+
+
+\part{Stochastic Calculus}
+\label{part:calculus}
+
 \input{./diffusion/index.tex}
-\input{./calculus/index.tex}
 \input{./sde/index.tex}
 \input{./mal/index.tex}
+\input{./calculus/index.tex}

src/mal/bridge.tex

@@ -0,0 +1,40 @@
+\section{The Brownian Bridge}
+\label{section:brownian-bridge}
+
+
+\begin{definition}[Brownian Bridge]
+\label{definition:brownian-bridge}
+    Let $(\Omega, \cf, \bp)$ be a probability space, $a \in \real^d$, and $X: \Omega \to C([0, 1]; \real^d)$ be a Gaussian process, then $X$ is a \textbf{Brownian bridge} from $0$ to $a$ such that:
+    \begin{enumerate}
+        \item For each $t \in [0, 1]$, $\ev(X_t) = at$.
+        \item For each $s, t \in [0, 1]$, $\text{Cov}(X_s, X_t) = (s \wedge t - st)I$. 
+    \end{enumerate}
+\end{definition}
+
+\begin{theorem}[{{\cite[Theorem 40.3]{Rogers}}}]
+\label{theorem:brownian-bridge}
+    Let $a \in \real^d$, then the following are equivalent definitions of the distribution of the Brownian bridge from $0$ to $a$. 
+    \begin{enumerate}
+        \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, 
+        \[
+        X_t = B_t + t(a - B_1)
+        \]
+        \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, 
+        \[
+        X_t = at + (1 - t)B_{t/(1-t)}
+        \]
+        \item $X$ is a continuous process such that
+        \[
+        Y_t = X_t - at + \int_0^1 \frac{X_s - as}{1 - s}ds
+        \]
+
+        is a Brownian motion. 
+        \item The distribution of $X$ is the $h$-transform of the classical Wiener measure on $C([0, 1]; \real^d)$, where
+        \[
+        h(t, x) = \frac{1}{[2\pi(1 - t)]^{d/2}}\exp\braks{-\frac{\norm{x - a}_{\real^d}^2}{2(1 - t)}}
+        \]
+    \end{enumerate}
+    
+\end{theorem}
+
+

src/mal/brownian.tex

@@ -1,5 +1,5 @@
-\section{$h$-Brownian Motions}
-\label{section:h-brownian}
+\section{Girsanov Transforms}
+\label{section:girsanov}
 
 \begin{theorem}[Cameron-Martin-Girsanov, {{\cite[Theorem 38.5]{Rogers}}}]
 \label{theorem:cameron-martin}
@@ -45,132 +45,3 @@
     \end{enumerate}
 \end{theorem}
 
-\begin{definition}[Mean Value Property]
-\label{definition:mean-value-property}
-    Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$. For each $0 \le s \le t \le T$ and $x \in \real^d$, let $P_{s, t}(x, \cdot)$ be its transition distribution, and $h: (0, \infty) \times \real^d \to [0, \infty)$ be a measurable function, then $h$ satisfies the \textbf{mean-value property} with respect to $\bp$ if for any $0 \le s \le t \le T$ and $x \in \real^d$, 
-    \[
-    h(s, x) = \int_{\real^d}h(t, y)P_{s, t}(x, dy)
-    \]
-\end{definition}
-
-
-\begin{lemma}
-\label{lemma:h-density-martingale}
-    Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, and $A_0 = \bracs{h(0, \cdot) > 0}$, then the process
-    \[
-    \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}
-    \]
-
-    is a $\bracs{\mathcal{F}_t}$-martingale with respect to $\bp$. 
-\end{lemma}
-\begin{proof}
-    Let $t \in [0, T]$ and $\mu$ be the initial distribution of $\bp$, then
-    \[
-        \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}} = 
-        \int_{A_0} \frac{1}{h(0, x)}\int_{\real^d} h(t, y)P_{0, t}(x, dy) \mu(dx) = \mu(A_0)
-    \]
-
-    so the given process is integrable. Now, since $h$ satisfies the mean-value property, 
-    \begin{align*}
-        \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(T, X_T)}{h(0, X_0)} \bigg | \cf_t} &= \frac{\one_{\bracs{X_0 \in A_0}}}{h(0, X_0)} \cdot \ev_\bp\braks{h(t, X_t)| \cf_t} \\
-        &= \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}
-    \end{align*}
-    
-    
-\end{proof}
-
-
-\begin{proposition}
-\label{proposition:space-time-regular-measure}
-    Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\mu$ be its initial distribution, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, then there exists a probaiblity measure $\bp^h$ on $D([0, \infty); \real^d)$ such that:
-     \begin{enumerate}
-         \item For each $t \ge 0$, 
-         \[
-         \frac{d \bp^h}{d \bp} \bigg |_{\cf_{t^+}} = \frac{\one_{\bracs{X_0 \in A_0}}}{\mu(A_0)}\frac{h(t, X_t)}{h(0, X_0)}
-         \]
-
-         where $A_0 = \bracs{h(0, \cdot) > 0}$. 
-
-         \item Under $\bp^h$, $X_t$ is a Markov process with transition function
-         \[
-         P_{s, t}^h(x, dy) = \begin{cases}
-            \frac{h(t, y)}{h(s, x)}P_{s, t}(x, dy) &h(s, x) > 0 \\
-            0 & h(s, x) = 0
-         \end{cases}
-         \]
-         \item If $\bp$ is the classical Wiener measure, then the process
-         \[
-         \tilde X_t = X_t - \int_0^t \frac{\partial_x h(s, X_s)}{h(s, X_s)}ds
-         \]
-
-         is a $\bp^h$-Brownian motion. 
-     \end{enumerate}
-
-     The distribution $\bp^h$ is the \textbf{$h$-transform} of $\mathbf{P}$.
-     
-\end{proposition}
-
-
-\begin{theorem}[Reuter, {{\cite[Theorem 39.66]{Rogers}}}]
-\label{theorem:reuter}
-    Let $(\Omega, \cf, \bp)$ be a filtered probability space, $X: \Omega \to C([0, \infty); \real^d)$ be a standard $\bracs{\mathcal{F}_t}$-Brownian motion with drift $\mu$ starting at $0$. If $\tau = \inf\bracs{t \ge 0: |X_t| = 1}$, then $X_\tau$ and $\tau$ are independent. 
-\end{theorem}
-\begin{proof}
-    If $\mu = 0$, then $X_\tau$ and $\tau$ are independent by rotational invariance. 
-    
-    Otherwise, let $\wien$ be the classical Wiener measure on $C([0, \infty); \real^d)$, $\mathcal{V}$ be the distribution of $X$, and $Y$ be the canonical process on $C([0, \infty); \real^d)$. For each $t \ge 0$, let $\mathcal{G}_t = \sigma(\bracs{Y_s|0 \le s \le t})$, then by the Cameron-Martin formula, 
-    \[
-    h(t, Y_t) = \frac{d \mathcal{V}}{d \wien} \bigg |_{\mathcal{G}_t} = \exp\braks{\angles{\mu, Y_t}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 t}
-    \]
-
-    Since $h(t \wedge \tau, X_{t \wedge \tau})$ is a uniformly integrable martingale, 
-    \[
-    \frac{d \mathcal{V}}{d\wien} \bigg |_{\mathcal{G}_{T^+}} = h(\tau, Y_\tau)
-    \]
-
-    In which case, for any measurable functions $f: \mathbb{S}^{d} \to [0, 1]$ and $g: [0, \infty) \to [0, 1]$, 
-    \begin{align*}
-        \ev^{\mathcal{V}}[f(Y_\tau)g(\tau)] &= \ev^{\wien}\braks{f(Y_\tau)g(\tau)\exp(\angles{\mu, Y_\tau}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau)} \\
-        &= \ev^\wien[f(Y_\tau)e^{\angles{\mu, Y_\tau}_{\real^d}}]\ev^\wien[g(\tau)e^{- \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau}] \\
-        &= \ev^{\mathcal{V}}(f(Y_\tau))\ev^{\mathcal{V}}(g(\tau))
-    \end{align*}
-\end{proof}
-
-
-\begin{definition}[Brownian Bridge]
-\label{definition:brownian-bridge}
-    Let $(\Omega, \cf, \bp)$ be a probability space, $a \in \real^d$, and $X: \Omega \to C([0, 1]; \real^d)$ be a Gaussian process, then $X$ is a \textbf{Brownian bridge} from $0$ to $a$ such that:
-    \begin{enumerate}
-        \item For each $t \in [0, 1]$, $\ev(X_t) = at$.
-        \item For each $s, t \in [0, 1]$, $\text{Cov}(X_s, X_t) = (s \wedge t - st)I$. 
-    \end{enumerate}
-\end{definition}
-
-\begin{theorem}[{{\cite[Theorem 40.3]{Rogers}}}]
-\label{theorem:brownian-bridge}
-    Let $a \in \real^d$, then the following are equivalent definitions of the distribution of the Brownian bridge from $0$ to $a$. 
-    \begin{enumerate}
-        \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, 
-        \[
-        X_t = B_t + t(a - B_1)
-        \]
-        \item There exists a Brownian motion $B$ such that for each $t \in [0, 1]$, 
-        \[
-        X_t = at + (1 - t)B_{t/(1-t)}
-        \]
-        \item $X$ is a continuous process such that
-        \[
-        Y_t = X_t - at + \int_0^1 \frac{X_s - as}{1 - s}ds
-        \]
-
-        is a Brownian motion. 
-        \item The distribution of $X$ is the $h$-transform of the classical Wiener measure on $C([0, 1]; \real^d)$, where
-        \[
-        h(t, x) = \frac{1}{[2\pi(1 - t)]^{d/2}}\exp\braks{-\frac{\norm{x - a}_{\real^d}^2}{2(1 - t)}}
-        \]
-    \end{enumerate}
-    
-\end{theorem}
-
-
-

src/mal/h.tex

@@ -0,0 +1,93 @@
+\section{$h$-Transforms}
+\label{section:h-transform}
+
+
+\begin{definition}[Mean Value Property]
+\label{definition:mean-value-property}
+    Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$. For each $0 \le s \le t \le T$ and $x \in \real^d$, let $P_{s, t}(x, \cdot)$ be its transition distribution, and $h: (0, \infty) \times \real^d \to [0, \infty)$ be a measurable function, then $h$ satisfies the \textbf{mean-value property} with respect to $\bp$ if for any $0 \le s \le t \le T$ and $x \in \real^d$, 
+    \[
+    h(s, x) = \int_{\real^d}h(t, y)P_{s, t}(x, dy)
+    \]
+\end{definition}
+
+
+\begin{lemma}
+\label{lemma:h-density-martingale}
+    Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, and $A_0 = \bracs{h(0, \cdot) > 0}$, then the process
+    \[
+    \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}
+    \]
+
+    is a $\bracs{\mathcal{F}_t}$-martingale with respect to $\bp$. 
+\end{lemma}
+\begin{proof}
+    Let $t \in [0, T]$ and $\mu$ be the initial distribution of $\bp$, then
+    \[
+        \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}} = 
+        \int_{A_0} \frac{1}{h(0, x)}\int_{\real^d} h(t, y)P_{0, t}(x, dy) \mu(dx) = \mu(A_0)
+    \]
+
+    so the given process is integrable. Now, since $h$ satisfies the mean-value property, 
+    \begin{align*}
+        \ev_\bp\braks{\one_{\bracs{X_0 \in A_0}} \frac{h(T, X_T)}{h(0, X_0)} \bigg | \cf_t} &= \frac{\one_{\bracs{X_0 \in A_0}}}{h(0, X_0)} \cdot \ev_\bp\braks{h(t, X_t)| \cf_t} \\
+        &= \one_{\bracs{X_0 \in A_0}} \frac{h(t, X_t)}{h(0, X_0)}
+    \end{align*}
+    
+    
+\end{proof}
+
+\begin{proposition}
+\label{proposition:space-time-regular-measure}
+    Let $\bp$ be the distribution of a strong Markov process on $D([0, T]; \real^d)$ with transition functions $P_{s, t}(x, \cdot)$, $\mu$ be its initial distribution, $\bracs{X_t|t \in [0, T]}$ be the canonical process, $\bracs{\cf_t|t \in [0, T]}$ be its natural filtration, $h: (0, \infty) \times \real^d \to [0, \infty)$ be a function satisfying the mean-value property with respect to $\bp$, then there exists a probaiblity measure $\bp^h$ on $D([0, \infty); \real^d)$ such that:
+     \begin{enumerate}
+         \item For each $t \ge 0$, 
+         \[
+         \frac{d \bp^h}{d \bp} \bigg |_{\cf_{t^+}} = \frac{\one_{\bracs{X_0 \in A_0}}}{\mu(A_0)}\frac{h(t, X_t)}{h(0, X_0)}
+         \]
+
+         where $A_0 = \bracs{h(0, \cdot) > 0}$. 
+
+         \item Under $\bp^h$, $X_t$ is a Markov process with transition function
+         \[
+         P_{s, t}^h(x, dy) = \begin{cases}
+            \frac{h(t, y)}{h(s, x)}P_{s, t}(x, dy) &h(s, x) > 0 \\
+            0 & h(s, x) = 0
+         \end{cases}
+         \]
+         \item If $\bp$ is the classical Wiener measure, then the process
+         \[
+         \tilde X_t = X_t - \int_0^t \frac{\partial_x h(s, X_s)}{h(s, X_s)}ds
+         \]
+
+         is a $\bp^h$-Brownian motion. 
+     \end{enumerate}
+
+     The distribution $\bp^h$ is the \textbf{$h$-transform} of $\mathbf{P}$.
+     
+\end{proposition}
+
+
+\begin{theorem}[Reuter, {{\cite[Theorem 39.66]{Rogers}}}]
+\label{theorem:reuter}
+    Let $(\Omega, \cf, \bp)$ be a filtered probability space, $X: \Omega \to C([0, \infty); \real^d)$ be a standard $\bracs{\mathcal{F}_t}$-Brownian motion with drift $\mu$ starting at $0$. If $\tau = \inf\bracs{t \ge 0: |X_t| = 1}$, then $X_\tau$ and $\tau$ are independent. 
+\end{theorem}
+\begin{proof}
+    If $\mu = 0$, then $X_\tau$ and $\tau$ are independent by rotational invariance. 
+    
+    Otherwise, let $\wien$ be the classical Wiener measure on $C([0, \infty); \real^d)$, $\mathcal{V}$ be the distribution of $X$, and $Y$ be the canonical process on $C([0, \infty); \real^d)$. For each $t \ge 0$, let $\mathcal{G}_t = \sigma(\bracs{Y_s|0 \le s \le t})$, then by the Cameron-Martin formula, 
+    \[
+    h(t, Y_t) = \frac{d \mathcal{V}}{d \wien} \bigg |_{\mathcal{G}_t} = \exp\braks{\angles{\mu, Y_t}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 t}
+    \]
+
+    Since $h(t \wedge \tau, X_{t \wedge \tau})$ is a uniformly integrable martingale, 
+    \[
+    \frac{d \mathcal{V}}{d\wien} \bigg |_{\mathcal{G}_{T^+}} = h(\tau, Y_\tau)
+    \]
+
+    In which case, for any measurable functions $f: \mathbb{S}^{d} \to [0, 1]$ and $g: [0, \infty) \to [0, 1]$, 
+    \begin{align*}
+        \ev^{\mathcal{V}}[f(Y_\tau)g(\tau)] &= \ev^{\wien}\braks{f(Y_\tau)g(\tau)\exp(\angles{\mu, Y_\tau}_{\real^d} - \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau)} \\
+        &= \ev^\wien[f(Y_\tau)e^{\angles{\mu, Y_\tau}_{\real^d}}]\ev^\wien[g(\tau)e^{- \frac{1}{2}\norm{\mu}_{\real^d}^2 \tau}] \\
+        &= \ev^{\mathcal{V}}(f(Y_\tau))\ev^{\mathcal{V}}(g(\tau))
+    \end{align*}
+\end{proof}

src/mal/index.tex

@@ -1,4 +1,6 @@
-\chapter{Unknown Chapter}
+\chapter{$h$-Transforms}
 \label{chap:mal}
 
 \input{./brownian.tex}
+\input{./h.tex}
+\input{./bridge.tex}