Cleanup

src/process/index.tex

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+\part{Stochastic Processes}
+\label{part:stochastic-processes}
+
+\input{./markov/index.tex}

src/process/markov/brownian.tex

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+\section{Brownian Motion}
+\label{section:brownian-markov}
+
+\begin{proposition}[{{\cite[Proposition 3.4]{Baudoin}}}]
+\label{proposition:brownian-markov}
+    Let $(\Omega, \cf, \bp)$ be a probability space, $\bracs{B_t|t \ge 0}$ be a $\real$-valued stochastic process with $B_0 = 0$, then the following are equivalent:
+    \begin{enumerate}
+        \item $\bracs{B_t|t \ge 0}$ is a standard Brownian motion.
+        \item $\bracs{B_t|t \ge 0}$ is a Markov process with semigroup
+        \[
+        \bp_0 = \text{Id} \quad
+        (\bp_t f)(x) = \frac{1}{\sqrt{2\pi t}}\int f(y) \exp\paren{-\frac{(x - y)^2}{2t}}dy
+        \]
+
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    Let $\bracs{\cf_t|t \ge 0}$ be the natural filtration of $\bracs{B_t|T \ge 0}$, then for any $s, t \ge 0$ and $\xi \in \real$,
+
+    By the Markov property,
+    \begin{align*}
+        \ev\braks{\exp\paren{i \xi B_{t + s}}|\cf_s} &= \frac{1}{\sqrt{2\pi t}}\int \exp\paren{i \xi y}\exp\paren{-\frac{(B_s - y)^2}{2t}}dy \\
+        &= \frac{1}{\sqrt{2\pi t}}\int \exp\paren{i \xi (B_s + y)}\exp\paren{-\frac{y^2}{2t}}dy \\
+        \ev\braks{\exp\paren{i \xi (B_{t + s} - B_s)}|\cf_s} &= \frac{1}{\sqrt{2\pi t}}\int \exp\paren{i \xi y}\exp\paren{-\frac{y^2}{2t}}dy = e^{-t\xi^2/2}
+    \end{align*}
+    So $\bracs{B_t|t \ge 0}$ admits independent and homogeneous increments, and is the standard Brownian motion.
+
+    Now suppose that $\bracs{B_t| t \ge 0}$ is the standard Brownian motion, then for any $s, t \ge 0$ and bounded Borel function $f: \real \to \real$,
+    \[
+    \ev[f(B_{t+s})|B_s = x] = \frac{1}{\sqrt{2\pi t}}\int f(x + y) \exp\paren{-\frac{y^2}{2t}} dy
+    \]
+
+    so
+    \[
+    \ev[f(B_{t + s})|\cf_s] = \int f(B_s + y)\exp\paren{-\frac{y^2}{2t}}dy
+    \]
+
+\end{proof}

src/process/markov/definition.tex

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+\section{Transition Functions}
+\label{section:markov-transition}
+
+\begin{definition}[Transition Function]
+\label{definition:transition-function}
+    Let $(X, \cm)$ be a measurable space. A \textbf{transition function} $\bracs{P_t| t \ge 0}$ on $X$ is a collection of maps
+    \[
+    P_t: X \times \cm \to [0, 1]
+    \]
+
+    such that:
+    \begin{enumerate}
+        \item For each $t \ge 0$ and $x \in X$, $P_t(x, \cdot)$ is a probability measure on $X$.
+        \item For each $t \ge 0$ and $A \in \cm$, $x \mapsto P_t(x, A)$ is $(\cm, \cb_\real)$-measurable.
+        \item For any $s, t \ge 0$, $x \in X$, and $A \in \cm$,
+        \[
+        P_{t + s}(x, A) = \int P_t(y, A)P_s(x, dy)
+        \]
+
+    \end{enumerate}
+\end{definition}
+
+\begin{definition}[Semigroup of Transition Function]
+\label{definition:transition-function-semigroup}
+    Let $(X, \cm)$ be a measurable space, $\bracs{P_t|t \ge 0}$ be a transition function on $X$. For each $f: X \to \complex$ bounded and $(\cm, \cb_\complex)$-measurable, let
+    \[
+    (\bp_t f)(x) = \int f(y)P_t(x, dy)
+    \]
+
+    then $\bracs{\bp_t|t \ge 0}$ is a semigroup such that:
+    \begin{enumerate}
+        \item For each $t \ge 0$, $\bp_t \one = \one$.
+        \item For each $t \ge 0$, $\bp_t$ is a positive linear functional.
+    \end{enumerate}
+    known as the \textbf{semigroup} of $\bracs{P_t|t \ge 0}$.
+\end{definition}
+\begin{proof}
+    Let $s, t \ge 0$, then for any $A \in \cm$,
+    \[
+    (\bp_{s+t}\one_A)(x) = \int \one_A(y)P_{s+t}(x, dy) = \int P_t(y, A)P_s(x, dy) = \int \int \one_A(z)P_t(y, dz)P_s(x, dy)
+    \]
+
+    Let $f: X \to \complex$ bounded and $(\cm, \cb_\complex)$-measurable. By \autoref{proposition:measurable-simple-separable-norm}, there exists $\seq{f_n} \subset \Sigma(X; \complex)$ such that $\abs{f_n} \le \abs{f}$ for all $n \in\nat$, and $f_n \to f$ pointwise. By the \hyperref[Dominated Convergence Theorem]{theorem:dct},
+    \begin{align*}
+    (\bp_{s+t}f)(x) &= \limv{n}\int f_n(y)P_{s+t}(x, dy) = \limv{n}\iint f_n(z)P_t(y, dz)P_s(x, dy) \\
+    &= \int \int f_n(z)P_t(y, dz)P_s(x, dy) = (\bp_s\bp_t) f(x)
+    \end{align*}
+\end{proof}
+
+\begin{definition}[Markov Process]
+\label{definition:markov-process}
+    Let $(\Omega, \cf, \bp)$ be a probability space, $(Y, \cm)$ be a measurable space, $\bracs{X_t|t \ge 0}$ be a stochastic process, and $\bracs{\cf_t|t \ge 0}$ be its natural filtration.
+
+    The process $\bracs{X_t|t \ge 0}$ is \textbf{Markov} if there exists a transition function $\bracs{P_t|t \ge 0}$ such that for every $f: Y \to \complex$ bounded and $(\cm, \cb_\complex)$-measurable,
+    \[
+    \ev\braks{f(X_{t+s})|\cf_s} = (\bp_tf)(X_s)
+    \]
+
+\end{definition}
+
+
+\begin{theorem}[{{\cite[Theorem 3.14]{Baudoin}}}]
+\label{theorem:markov-existence}
+    Let $(Y, \cm)$ be a measurable space, $\nu: \cm \to [0, 1]$ be a probability measure on $Y$, and $\bracs{P_t|t \ge 0}$ be a transition function on $Y$, then there exists a probability space $(\Omega, \cf, \bp)$ and a stochastic process $\bracs{X_t|t \ge 0}$ such that:
+    \begin{enumerate}
+        \item The distribution of $X_0$ is $\nu$.
+        \item $\bracs{X_t|t \ge 0}$ is a Markov process associated with $\bracs{P_t|t \ge 0}$.
+    \end{enumerate}
+\end{theorem}
+\begin{proof}
+    For each $N \in \natp$, $A \in \cm$ and $B \in \bigotimes_{n = 1}^N\cm$, and $0 = t_0 < t_1 < \cdots < t_N$, let
+    \[
+    \mu_{t_0, t_1, \cdots, t_N} = \int_A\int_B P_{t_1}(z, dx_1)P_{t_2 - t_1}(x_1, dx_2) \cdots P_{t_N - t_{N-1}}(x_{N_1}, dx_N) \nu(dz)
+    \]
+
+    then $\mu_{t_0, t_1, \cdots, t_N}$ is a probability measure on $\cm \times $
+\end{proof}

src/process/markov/index.tex

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+\chapter{Markov Processes}
+\label{chap:markov}
+
+
+\input{./definition.tex}
+\input{./brownian.tex}