Minor update

preamble.sty

@@ -167,6 +167,7 @@
 \newcommand{\cd}{\mathcal{D}}
 \newcommand{\calr}{\mathcal{R}}
 \newcommand{\scp}{\mathscr{P}}
+\newcommand{\wien}{\mathcal{W}}
 
 % Jokes
 \newcommand{\lol}{\boxed{\text{LOL.}}}

spec.toml

@@ -16,6 +16,8 @@ primaryColour = "blue"
 neutralColour = "grey"
 searchLimit = 16
 maxSearchPages = 48
-recentChanges = 10
+recentChanges = 0
-tableOfContentsDepth = 2
+tableOfContentsDepth = 1
+hoverPreview = false
+copyLabelButton = false
 advertiseSpec = true

src/sde/exact.tex

@@ -100,4 +100,50 @@
 \end{proof}
 
 
+\begin{lemma}[Gronwall]
+\label{lemma:gronwall}
+    Let $f \in C([0, T]; \real)$, and $c, K > 0$ such that
+    \[
+    f(t) \le c + K\int_0^t f(s)ds
+    \]
+
+    for all $t \in [0, T]$, then
+    \[
+    \frac{d}{dt}\braks{e^{-Kt}\int_0^t f(s)ds} \le ce^{-Kt}
+    \]
+
+    and $f(t) \le ce^{Kt}$. 
+\end{lemma}
+
+
+\begin{theorem}[Blagoveshchenskii-Blagoveshchensk]
+\label{theorem:strong-solution-properties}
+    Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals satisfying the \hyperref[Lipschitz condition]{definition:lipschitz-coefficient}. If for each $T \ge 0$, 
+    \[
+    \sup_{0 \le s \le T}\norm{\sigma(s, 0)}_{L(\real^d; \real^n)} + \norm{b(s, 0)}_{\real^n} < \infty
+    \]
+
+    then the SDE
+    \[
+    X_t = \xi + \int_0^t \sigma(s, X)dB_s + \int_0^t b(s, X)ds
+    \]
+
+    admits a strong solution
+    \[
+    F: \real^n \times C([0, \infty); \real^d) \to C([0, \infty); \real^n)
+    \]
+
+    such that:
+    \begin{enumerate}
+        \item For each $\theta \in C([0, \infty); \real^d)$, $F(\cdot, \theta): \real^d \to C([0, \infty); \real^n)$ is continuous.
+        \item For each solution $X^y$ initial condition $y \in \real^n$ and $s \ge 0$, 
+        \[
+        X_{s + t}^y = F(X_s^y, \tau_{-s}B)_t \quad \forall t \ge 0
+        \]
+
+        almost surely, where $(\tau_{-s}B)_r = B_{r+s}$. 
+        \item For each $y \in \real^n$, $\bracs{X_t^y|t \ge 0}$ is a Markov process. 
+    \end{enumerate}
+    
+\end{theorem}
 

src/sde/index.tex

@@ -3,4 +3,6 @@
 
 \input{./setup}
 \input{./exact}
+\input{./weak}
+\input{./martingale}
 

src/sde/martingale.tex

@@ -0,0 +1,81 @@
+\section{The Martingale Formulation}
+\label{section:weak-martingale}
+
+
+\begin{definition}[Martingale Problem]
+\label{definition:martingale-problem}
+    Let $a: [0, \infty) \times C([0, \infty); \real^n) \to L(\real^n; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^n) \to \real^n$ be previsible path functionals. For each $u \in C_c^\infty(\real^n)$, let
+    \[
+    Lu = \frac{1}{2}\dpn{A, D^2f}{\real^{n \times n}} + \dpn{b, Df}{\real^n}
+    \]
+
+    then for any $y \in \real^n$, filtered probability space $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$, and $X: \Omega \to C([0, \infty); \real^n)$, $X$ is a \textbf{solution to the martingale problem for $(a, b)$ starting at $y$} if:
+    \begin{enumerate}
+        \item $X_0 = y$ almost surely. 
+        \item For each $f \in C_c^\infty(\real^n)$, the process
+        \[
+        C_t^f = f(X_t) - f(X_0) - \int_0^t Lf(s, X_s)ds
+        \]
+
+        is a $\bracs{\mathcal{F}_t}$-martinagle. 
+    \end{enumerate}
+
+    If the distribution of $X$ on $C([0, \infty); \real^n)$ is the unique distribution satisfying the above, then the solution for the martingale problem is unique. If for each $y \in \real^n$, such a solution exists, then the martingale problem is well posed. 
+\end{definition}
+
+\begin{theorem}[Equivalence of Formulations]
+\label{theorem:equivalence-of-formulations}
+    Let $(\Omega, \bracsn{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-Brownian motion, $\sigma: [0, \infty) \times C([0, \infty); \real^n) \to L(\real^n; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^n) \to \real^n$ be bounded measurable functions such that $\sigma_t$ is invertible for all $t \ge 0$. 
+    
+    Let $y \in \real^n$ and $X: \Omega \to C([0, \infty); \real^n)$ be a solution to the martingale problem for $(\sigma^*\sigma, b)$ starting at $y$, then there exists a weak solution of the SDE
+    \[
+    Y_t = Y_0 + \int_0^t \sigma(s, Y) dB_s + \int_0^t b(s, Y)ds
+    \]
+
+    starting at $y$ whose distribution is the same as $X$. 
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem 20.1]{Rogers}}}. ]
+    By truncation, for each $1 \le i \le n$,
+    \[
+    M_t = X_t - \int_0^t b(s, X)ds
+    \]
+
+    and
+    \[
+    M_tM_t^T - \int_0^t a(s, X)ds
+    \]
+
+    are local martingales. In which case, by Lévy's characterisation of Brownian motion, 
+    \[
+    \td B_t = \int_0^t \sigma(s, X)^{-1}dM_s
+    \]
+
+    is a standard Brownian motion, and $X$ and $\td B$ satisfy the SDE.
+\end{proof}
+
+\begin{theorem}
+\label{theorem:martingale-markov}
+    Let $a: \real^n \to L(\real^n; \real^n)$ and $b: \real^n \to \real^n$ be bounded measurable functions such that the martingale problem for $(a, b)$ is well-posed, that is, let
+    \[
+    Lu = \frac{1}{2}\dpn{A, D^2f}{\real^{n \times n}} + \dpn{b, Df}{\real^n}
+    \]
+    
+    then for each $y \in \real^n$, there exists a unique probability measure $\bp^y$ on $C([0, \infty); \real^n)$ such that:
+    \begin{enumerate}
+        \item $\bp^y(x_0 = y) = 1$. 
+        \item For each $f \in C_c^\infty$, 
+        \[
+        C_t^f = f(\pi_t) - f(x_0) - \int_0^t Lf(x_s)ds
+        \]
+
+        is a $\bp^y$-martingale. 
+    \end{enumerate}
+    
+    then 
+    \begin{enumerate}
+        \item $\bracs{x_t|t \ge 0}$ is a time-homogeneous strong Markov process with respect to $\bp^y$.
+        \item If $a = \sigma \sigma^*$ and $(\sigma, b)$ satisfy the \hyperref[Lipschitz condition]{definition:lipschitz-coefficient}, then the generator of the above process is $L$.
+    \end{enumerate}
+\end{theorem}
+
+

src/sde/setup.tex

@@ -21,6 +21,12 @@
     and $\mathscr{J}_{C([0, \infty); \real^d)}$ be the previsible $\sigma$-algebra on $(0, \infty) \times C([0, \infty); \real^d)$, then a \textbf{previsible path functional} is a $\mathscr{J}$-measurable mapping on $(0, \infty) \times C([0, \infty); \real^d)$. 
 \end{definition}
 
+\begin{definition}[Augmentation]
+\label{definition:augmentation}
+    Let $(\Omega, \bracs{\cf_t}, \bp)$ be a filtered probability space and $\mathcal{N}$ be the collection of all $\bp$-null sets in $\cf = \sigma(\bracs{\cf_t|t \ge 0})$. For each $t \ge 0$, let $\ol{\cf}_t = \sigma(\cf_t \cup \mathcal{N})$, then the filtration $\bracsn{\ol{\cf}_t|t \ge 0}$ is the \textbf{$\bp$-augmentation} of $\bracs{\mathcal{F}_t}$. 
+\end{definition}
+
+
 \begin{lemma}
 \label{lemma:adapted-composition}
     Let $(\Omega, \bracs{\cf_t})$ be a filtered space and $X: \Omega \to C([0, \infty); \real^d)$ be a $\bracs{\mathcal{F}_t}$-adapted process with continuous sample paths, then
@@ -51,9 +57,9 @@
 
 \begin{definition}[Pathwise Uniqueness]
 \label{definition:pathwise-uniqueness}
-    Let $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then the diffusion SDE
+    Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals, then the SDE
     \begin{equation}
-    X_t = \xi + \int_0^t \sigma(X_s) dB_s + \int_0^t b(X_s)ds \label{equation:diffusion-sde}
+    X_t = \xi + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds\label{equation:diffusion-sde}
     \end{equation}
 
     has \textbf{pathwise uniqueness} if given
@@ -68,10 +74,10 @@
 
 \begin{definition}[Pathwise Exact]
 \label{definition:pathwise-exact}
-    Let $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then the diffusion SDE
+    Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals, then the SDE
-    \begin{equation}
+    \[
-    X_t = \xi + \int_0^t \sigma(X_s) dB_s + \int_0^t b(X_s)ds \label{equation:diffusion-sde}
+    X_t = \xi + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds
-    \end{equation}
+    \]
 
     is \textbf{pathwise exact} if given
     \begin{itemize}
@@ -86,5 +92,37 @@
 
 
 
+\begin{definition}[Strong Solution]
+\label{definition:sde-strong-solution}
+    Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals, then a mapping
+    \[
+    F: \real^n \times C([0, \infty); \real^d) \to C([0, \infty); \real^n)
+    \]
+
+    is a \textbf{strong solution} to the SDE
+    \begin{equation}
+    \label{eq:sde-strong-solution}
+    X_t = \xi + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds
+    \end{equation}
+
+    if
+    \begin{enumerate}
+        \item For each $t \ge 0$, let
+        \[
+        \mathcal{F}_t = \sigma(\bracs{\pi_s: C([0, \infty); \real^d) \to \real^d|0 \le s \le t}) \quad \mathcal{G}_t = \sigma(\bracs{\pi_s: C([0, \infty); \real^n) \to \real^n|0 \le s \le t})
+        \]
+
+        Let $\wien^d$ and $\wien^n$ be the classical Wiener measure on $C([0, \infty); \real^d)$ and $C([0, \infty); \real^n)$, respectively, and $\bracsn{\ol{\mathcal{G}}_t|t \ge 0}$ be the $\wien^n$-augmentation of $\bracs{\mathcal{G}_t|t \ge 0}$, then
+        \[
+        F^{-1}(\mathcal{F}_t) \subset \mathcal{B}(\real^n) \times \overline{\mathcal{G}_t}
+        \]
+
+        for all $t \ge 0$. 
+
+        \item For any filtered probability space $(\Omega, \bracs{\mathcal{H}_t}, \bp)$, random variable $\xi: \Omega \to \real^n$, and $\bracs{\mathcal{H}_t}$-Brownian motion $B$, the process $X = F(\xi, B)$ solves \autoref{eq:sde-strong-solution}.
+    \end{enumerate}
+    
+\end{definition}
+
 
 

src/sde/weak.tex

@@ -0,0 +1,54 @@
+\section{Weak Solutions}
+\label{section:weak-solution}
+
+
+\begin{definition}[Weak Solution]
+\label{definition:sde-weak-solution}
+    Let $\mu$ be a Borel probability measure on $\real^n$, $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals, then the SDE
+    \[
+    X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds
+    \]
+
+    has a \textbf{weak solution with initial distribution} $\mu$ if there exists a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$, a $\bracs{\mathcal{F}_t}$-Brownian motion $B$, and a $\bracs{\mathcal{F}_t}$-semimartingale $X: \Omega \to C([0, \infty); \real^d)$ such that:
+    \begin{enumerate}
+        \item $X_0$ has distribution $\mu$.
+        \item For each $t > 0$, 
+        \[
+        \int_0^t \norm{\sigma(s, X)}_{\real^n}^2 + \norm{\sigma(s, X)}_{\real^n}^2 ds < \infty
+        \]
+
+        almost surely.
+        \item For each $t \ge 0$,
+        \[
+        X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds
+        \]
+    \end{enumerate}
+\end{definition}
+
+\begin{definition}[Uniqueness in Distribution]
+\label{definition:uniqueness-in-distribution}
+    Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals, then the SDE
+    \[
+    X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds
+    \]
+
+    has \textbf{uniqueness in law} if for any solutions $X$ and $X'$ such that the distributions of $X_0$ and $X_0'$ are the same, the distributions of $X$ and $X'$ are the same. 
+\end{definition}
+
+\begin{theorem}[Yamada and Watanabe, {{\cite[Theorem 17.1]{Rogers}}}]
+\label{theorem:yamada-watanabe}
+    Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to L(\real^d; \real^n)$ and $b: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$ be previsible path functionals, and
+    \[
+    X_t = X_0 + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds
+    \]
+
+    then the above SDE is exact if and only if the following conditions hold:
+    \begin{enumerate}
+        \item The SDE has a weak solution.
+        \item The SDE has the pathwise-uniqueness property. 
+    \end{enumerate}
+    
+    If the above holds, then the SDE also has uniqueness in law. 
+\end{theorem}
+
+