Minor update

.vscode/project.code-snippets

@@ -126,6 +126,11 @@
     "prefix": "cal",
     "body": ["\\mathcal{$1}$0"]
   },
+  "Mathscr": {
+  "scope": "latex",
+    "prefix": "scr",
+    "body": ["\\mathscr{$1}$0"]
+  },
   "Mathfrak": {
   "scope": "latex",
     "prefix": "fk",
@@ -160,5 +165,10 @@
   "scope": "latex",
     "prefix": "filt",
     "body": ["$\\bracs{\\mathcal{F}_t}$"]
+  },
+  "Path Space": {
+    "scope": "latex",
+    "prefix": "path",
+    "body": ["C([0, \\infty); \\real^d)"]
   }
 }

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.}}}

refs.bib

@@ -8,3 +8,32 @@
   year={1997},
   publisher={Springer Berlin Heidelberg}
 }
+
+@book{Baudoin,
+  title={Diffusion Processes and Stochastic Calculus},
+  author={Baudoin, F.},
+  isbn={9783037191330},
+  lccn={2014395958},
+  series={EMS textbooks in mathematics},
+  url={https://books.google.ca/books?id=ov4kcKkzTP4C},
+  year={2014},
+  publisher={European Mathematical Society}
+}
+
+@book{Lang,
+  title={Differential and Riemannian manifolds},
+  author={Lang, Serge},
+  year={2012},
+  publisher={Springer Science \& Business Media}
+}
+
+@article{Rogers,
+  title={Book review: Diffusions, markov processes. and martingales, volume 2, it6 calculus},
+  author={Rogers, LCG and Williams, David},
+  journal={Stochastics: An International Journal of Probability and Stochastic Processes},
+  volume={27},
+  number={1},
+  pages={59--63},
+  year={1989},
+  publisher={Taylor \& Francis}
+}

spec.toml

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

src/calculus/frobenius.tex

@@ -0,0 +1,77 @@
+\section{Frobenius' Theorem}
+\label{section:frobenius}
+
+\begin{theorem}[Frobenius]
+\label{theorem:frobenius}
+    Let $X$ be a $C^p$ ($p \ge 2$) manifold and $E \subset TX$ be a subbundle, then the following are equivalent:
+    \begin{enumerate}
+        \item For each pair of vector fields $\xi, \eta: X \to E$, $[\xi, \eta]$ lies in $E$. 
+        \item For each $\omega \in \Lambda^1(TX)$ vanishing on $E$, $\xi, \eta: X \to E$, $d\omega(\xi, \eta) = 0$. 
+        \item $E$ is integrable. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Section VI.1]{Lang}}}. ]
+    Let $\omega \in \Lambda^1(TX)$, then
+    \[
+    d\omega(\xi, \eta) = \omega([\xi, \eta]) - \eta\omega(\xi) - \xi\omega(\eta)
+    \]
+
+    (1) $\Rightarrow$ (2): If $\omega$ vanishes on $E$ and $[\xi, \eta]$ lies in $E$, then $d\omega$ vanishes on $(\xi, \eta)$. 
+
+    (2) $\Rightarrow$ (1): If $[\xi, \eta]$ does not lie in $E$, then there exists $\omega \in \Lambda^1(TX)$ that vanishes on $E$ but not $[\xi, \eta]$, which contradicts (2).
+
+    (1) $\Leftrightarrow$ (3): For any point on $X$, since $E$ is a subbundle, there exists Banach spaces $F, G$ and a coordinate neighbourhood $U \times V$ such that $E$ is given by a $C^{p-1}$-mapping
+    \[
+    f: U \times V \times F \to U \times V \times (F \times G)
+    \]
+
+    such that $\pi_1f(x, y)(z) = z$ for all $(x, y) \in U \times V$ and $z \in F$. Let
+    \[
+    g: U \times V \to L(F; G) \quad g(x, y)(z) = \pi_2f(x, y, z)
+    \]
+
+    then $g$ is also a $C^{p - 1}$-mapping. Let
+    \[
+    \Xi: U \times V \to F \times G
+    \]
+
+    be the local representation of a vector field, then the vector field lies in $E$ if and only if
+    \[
+    \pi_2\Xi(x, y) = g(x, y)\pi_1\Xi(x, y)
+    \]
+
+    for all $(x, y) \im U \times V$. Let $H: U \times V \to F \times G$ be another vector field, then by assumption (1), $[\Xi, H]$ lies in $E$. Denote $\xi = \pi_1\Xi$ and $\eta = \pi_1 H$, then
+    \begin{align*}
+        [\xi, \eta] &= D\eta \cdot \xi - D\xi \cdot \eta \\
+        g \cdot (D\eta \cdot \xi - D\xi \cdot \eta) &= (Dg \cdot \xi) \cdot \eta + g \cdot D\eta \cdot \xi - (Dg \cdot \eta) \cdot \xi - g \cdot D\eta \cdot \xi \\
+        (Dg \cdot \xi) \cdot \eta &= (Dg \cdot \eta) \cdot \xi
+    \end{align*}
+
+    Let $(x_0, y_0) \in U \times V$, then by \autoref{theorem:local-frobenius}, there exists $U_0 \in \cn_F(x_0)$ and $V_0 \in \cn_G(y_0)$ and $\alpha \in C^{p-1}(U_0 \times V_0; V)$ such that
+    \[
+    \partial_x\alpha(x, y) = g(x, \alpha(x, y))
+    \]
+
+    Let
+    \[
+    \varphi: U_0 \times V_0 \to U \times V \quad (x, y) \mapsto (x, \alpha(x, y))
+    \]
+
+    then
+    \[
+    D\varphi(x_0, y_0) = \begin{bmatrix} Id & 0 \\ g(x, \alpha(x, y)) & Id \end{bmatrix}
+    \]
+
+    so $\varphi$ is a local diffeomorphism at $(x_0, y_0)$. Since for any $(u, v) \in F \times G$, 
+    \[
+    \partial_x\varphi(x, y) \cdot (u, v) = (u, g(x, \alpha(x, y)) \cdot u)
+    \]
+
+    the bundle $E$ is integrable. 
+
+
+\end{proof}
+
+
+
+

src/calculus/index.tex

@@ -0,0 +1,5 @@
+\chapter{Integral Manifolds/Shenanigans}
+\label{chap:integral}
+
+\input{./integrable.tex}
+\input{./frobenius.tex}

src/calculus/integrable.tex

@@ -0,0 +1,147 @@
+\section{Integrable Bundles}
+\label{section:integrable}
+
+\begin{definition}[Integrable]
+\label{definition:integrable}
+    Let $X$ be a $C^p$ ($p \ge 2$) manifold, $E \subset TX$ be a subbundle, and $x_0 \in X$, then $E$ is \textbf{integrable at $x_0$} if there exists a submanifold $x_0 \in Y \subset X$ such that
+    \[
+    T\iota: TY \to E \subset TX
+    \]
+
+    is an isomorphism. If $E$ is integrable at every $x_0 \in X$, then $E$ is \textbf{integrable}.
+\end{definition}
+
+\begin{lemma}
+\label{lemma:split-ode}
+    Let $E, F$ be Banach spaces, $U \subset E$, $V \subset F$ be open sets, $\eps > 0$, $g \in C^k((-\eps, \eps) \times U \times V; F)$ ($k \ge 2$), and $(x_0, y_0) \in U \times V$, then there exists $\delta > 0$, $U_0 \in \cn_E(x_0)$, $V_0 \in \cn_F(x_0)$ and a unique $C^k$-mapping
+    \[
+    \beta: (-\delta, \delta) \times U_0 \times V_0 \to V
+    \]
+
+    such that
+    \begin{enumerate}
+        \item For all $(x, y) \in U_0 \times V_0$, $\beta(0, x, y) = y$.
+        \item For all $(t, x, y) \in (-\delta, \delta) \times U_0 \times V_0$, 
+        \[
+        \frac{d}{dt}\beta(t, x, y) = g(t, x, \beta(t, x, y))
+        \]
+        \item For each fixed $y \in V_0$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_0$, 
+        \[
+        (\partial_t\partial_x\beta)(t, x) = (\partial_xg)(t, x, \beta(t, x)) + (\partial_yg)(t, x, \beta(t, x)) \circ (\partial_x\beta)(t, x)
+        \]
+    \end{enumerate}
+\end{lemma}
+\begin{proof}[Proof, {{\cite[Proposition VI.2.1]{Lang}}}.]
+    Let
+    \[
+    G: (-\eps, \eps) \times U \times V \to E \times F \quad (t, x, y) \mapsto (0, g(t, x, y))
+    \]
+
+    then by the existence and uniqueness of ODEs, there exists $\delta > 0$, $U_0 \in \cn_E(x_0)$, $V_0 \in \cn_F(y_0)$, and a unique $C^k$-mapping
+    \[
+    B: (-\delta, \delta) \times U_0 \times V_0 \to E \times F
+    \]
+
+    such that
+    \begin{enumerate}
+        \item[(a)] For each $(x, y) \in U_0 \times V_0$, $B(0, x, y) = (x, y)$.
+        \item[(b)] For each $(t, x, y) \in (-\delta, \delta) \times U_0 \times V_0$, 
+        \[
+        \frac{d}{dt}B(t, x, y) = (0, g(t, B(t, x, y)))
+        \]
+    \end{enumerate}
+
+    Let $\beta(t, x, y) = \pi_2(t, x, y)$, then
+    \begin{enumerate}
+        \item For each $(x, y) \in U_0 \times V_0$, $\beta(0, x, y) = \pi_2(x, y) = y$. 
+        \item For each $(t, x, y) \in (-\delta, \delta) \times U_0 \times V_0$,
+        \[
+        \frac{d}{dt}\beta(t, x, y) = \pi_2(g(t, B(t, x, y))) = g(t, x, \beta(t, x, y))
+        \]
+        \item For each fixed $y \in V_0$, let $\beta(t, x) = \beta(t, x, y)$, then for each $(t, x) \in (-\delta, \delta) \times U_0$, by the chain rule, 
+        \[
+        (\partial_t\partial_x\beta)(t, x) = (\partial_xg)(t, x, \beta(t, x)) + (\partial_yg)(t, x, \beta(t, x)) \circ (\partial_x\beta)(t, x)
+        \]
+        
+    \end{enumerate}
+\end{proof}
+
+\begin{theorem}
+\label{theorem:local-frobenius}
+    Let $E, F$ be Banach spaces, $U \subset E$ and $V \subset F$ be open subsets, and $f \in C^k(U \times V; L(E; F))$ ($k \ge 2$). 
+    
+    For any $\xi, \eta \in C^k(U \times V; E)$, denote
+    \[
+    \Xi(x, y) = (\xi(x, y), f(x, y) \cdot \xi(x, y)) \quad H(x, y) = (\eta(x, y), f(x, y) \cdot \eta(x, y))
+    \]
+
+    If for every $\xi, \eta \in C^k(U \times V; E)$ and $(x, y) \in U \times V$,
+    \[
+    [Df(x, y) \cdot \Xi(x, y)] \cdot \eta(x, y) = [Df(x, y) \cdot H(x, y)] \cdot \xi(x, y)
+    \]
+
+    then for every $(x_0, y_0) \in U \times V$, there exists $U_0 \in \cn_E(x_0)$, $V_0 \in \cn_F(y_0)$, and a unique $\alpha \in C^k(U_0 \times V_0; V)$ such that
+    \begin{enumerate}
+        \item For every $y \in V_0$, $\alpha(x_0, y) = y$.
+        \item For every $(x, y) \in U_0 \times V_0$, 
+        \[
+        (\partial_x \alpha)(x, y) = f(x, \alpha(x, y))
+        \]
+    \end{enumerate}
+\end{theorem}
+\begin{proof}
+    Using translation, assume without loss of generality that $x_0 = 0$ and $y_0 = 0$. Let $B \in \cn_E(0)$ and
+    \[
+    g: (-\eps, \eps) \times B \times V \to F \quad (t, z, y) \mapsto f(tz, y) \cdot z
+    \]
+
+    then by \autoref{lemma:split-ode}, there exists $\delta > 0$, $B_0 \in \cn_E(0)$, and $\beta: (-\delta, \delta) \times B_0 \times V_0$ such that
+    \begin{enumerate}
+        \item[(a)] For each $(z, y) \in B_0 \times V_0$, $\beta(0, z, y) = y$. 
+        \item[(b)] For each $(t, z, y) \in (-\delta, \delta) \times B_0 \times V_0$, 
+        \[
+        \frac{d}{dt}\beta(t, z, y) = f(tz, \beta(t, z, y)) \cdot z
+        \]
+
+        \item[(c)] For each fixed $y \in V_0$, if $\beta(t, z) = \beta(t, z, y)$, then for each $(t, x) \in (-\delta, \delta) \times B_0$ and $h \in E$, 
+        \begin{align*}
+        (\partial_t\partial_z\beta)(t, z)\cdot h &= t(\partial_xf)(tz, \beta(t, z))\cdot h \cdot z
+        + f(tz, \beta(t, z))\cdot h \\
+        &+ (\partial_yf)(tz, \beta(t, z)) \circ (\partial_z \beta)(t, z)\cdot h \cdot z
+        \end{align*}
+    \end{enumerate}
+    
+    Following (c), let $k(t, z) = (\partial_z\beta)(t, z)\cdot h - tf(tz, \beta(t, z))\cdot h$, then
+    \begin{align*}
+        \frac{d}{dt}k(t) &= (\partial_t\partial_z\beta)(t, z)\cdot h - f(tz, \beta)\cdot h \\
+        &- t [(\partial_xf)(tz, \beta)]\cdot z \cdot h - t[(\partial_y f)(tz, \beta)]f(tz, \beta) \cdot z\cdot h
+    \end{align*}
+
+    where by assumption, 
+    \begin{align*}
+        &[(\partial_xf)(tz, \beta)]\cdot z \cdot h - [(\partial_y f)(tz, \beta)]f(tz, \beta) \cdot z \cdot h \\
+        &= Df(tz, \beta) \cdot (z, f(tz, \beta) \cdot z) \cdot h \\
+        &= Df(tz, \beta) \cdot (h, f(tz, \beta) \cdot h) \cdot z \\
+        &= [(\partial_xf)(tz, \beta)]\cdot h \cdot z - [(\partial_y f)(tz, \beta)]f(tz, \beta) \cdot h\cdot z
+    \end{align*}
+
+    Therefore by (c), 
+    \begin{align*}
+        \frac{d}{dt}k(t) &= (\partial_yf)(tz, \beta) \circ (\partial_z \beta)(t, z)\cdot h \cdot z - t[(\partial_y f)(tz, \beta)]f(tz, \beta) \cdot h\cdot z \\
+        &= (\partial_y f)(tz, \beta)[\partial_z \beta(t, z) - tf(tz, \beta)] \cdot h \cdot z \\
+        &= (\partial_y f)(tz, \beta) \cdot k(t) \cdot z
+    \end{align*}
+
+    Since $0$ is a solution to the above equation, $k(t) = 0$ by the uniqueness of solutions to ODEs. Hence for every $t \in (-\delta, \delta)$ and $z \in B_0$, 
+    \[
+    \partial_z \beta(t, z) = tf(tz, \beta(t, z))
+    \]
+
+    By adjusting $\delta$ and $B_0$, assume without loss of generality that $\delta > 1$. In which case, if $\alpha(x) = \beta(1, x, y)$, then
+    \[
+    \partial_x\alpha(x, y) = \partial_z\beta(1, x, y) = f(z, \beta(1, x, y)) = f(x, \alpha(x, y))
+    \]
+\end{proof}
+
+
+

src/diffusion/martingale.tex

@@ -30,6 +30,67 @@
     is a progressively measurable process. 
 \end{lemma}
 
+
+\begin{proposition}
+\label{proposition:martingale-quadratic}
+    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, $\bracs{X_t|t \ge 0}$ be a $\bracs{\mathcal{F}_t}$-progressively measurable process with continuous sample paths, and $f \in C_b^{1, 2}([0, \infty) \times \real^d)$. If the processes
+    \[
+    Y_t = f(t, X_t) - \int_0^t [(\partial_r + L_r)f](r, X_r)dr
+    \]
+    and
+    \[
+    W_t = f(t, X_t)^2 - \int_0^t [(\partial_r + L_r)(f^2)](r, X_r)dr
+    \]
+
+    are martingales, then
+    \[
+    [X]_t = \frac{1}{2}\int_0^t \dpn{Df, A(r)(Df)}{\real^d}(r, X_r)dr
+    \]
+
+\end{proposition}
+\begin{proof}
+    Firstly, since $f$ admits bounded derivatives, the integral term is of locally bounded variation. Therefore if $Z_t = f(t, X_t)$, then $[X]_t = \angles{Z}_t$. 
+
+    Let $\bracs{t_j}_0^n \subset [0, t]$ be a partition, then
+    \[
+        Z_t^2 - Z_0^2 = \sum_{j = 1}^{n}Z_{t_j}^2 - Z_{t_{j-1}}^2 = \sum_{j = 1}^n 2Z_{t_{j-1}}(Z_{t_j} - Z_{t_{j-1}}) + \sum_{j = 1}^n (Z_{t_j} - Z_{t_{j-1}})^2
+    \]
+
+    As the above holds for every partition, 
+    \[
+    Z_t^2 - Z_0^2 = 2\int_0^t Z_rZ(dr) + \angles{Z}_t
+    \]
+
+    Since $Z_t = Y_t + \int_0^t(\partial_r + L_r)fdr$, 
+    \begin{align*}
+        \int_0^t Z_rZ(dr) &= \int_0^t Z_r Y(dr) + \int_0^t f(r, X_r)(\partial_r + L_r)f(r, X_r)dr \\
+        Z_t^2 - Z_0^2 &= 2\int_0^t Z_r Y(dr) + 2\int_0^t f(r, X_r)(\partial_r + L_r)f(r, X_r)dr + \angles{Z}_t
+    \end{align*}
+
+    On the other hand, 
+    \[
+    Z_t^2 - Z_0^2 = W_t - W_0 + \int_0^t [(\partial_r + L_r)(f^2)](r, X_r)dr
+    \]
+
+    Therefore
+    \begin{align*}
+        \angles{Z}_t &= W_t - W_0 + \int_0^t [(\partial_r + L_r)(f^2)](r, X_r)dr \\
+        &- 2\int_0^t Z_r Y(dr) - 2\int_0^t f(r, X_r)(\partial_r + L_r)f(r, X_r)dr \\
+        &= W_t - W_0 - 2 \int_0^t f(r, X_r) Y(dr) \\
+        &+ \int_0^t [L_r(f^2)(r, X_r) - 2f(r, X_r)(L_rf)(r, X_r)]dr \\
+        &= W_t - W_0 - 2\int_0^t f(r, X_r)Y(dr) + \frac{1}{2}\int_0^t \dpn{Df, A(r)(Df)}{\real^d}(r, X_r)dr
+    \end{align*}
+
+    where the process $t \mapsto W_t - W_0 - 2\int_0^t f(r, X_r)Y(dr)$ is of finite variation, so it must be $0$ by \cite[Lemma 5.13]{Baudoin}. 
+    
+\end{proof}
+
+
 \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:
@@ -81,7 +142,7 @@
         \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
+            &\cdot \exp\braks{-\int_0^t [(\partial_r + L_r)g](r, X_r)}dr
         \end{align*}
 
         is a $\bracs{\mathcal{F}_t}$-martingale.
@@ -180,14 +241,6 @@
     \]
 
     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}
 
 

src/index.tex

@@ -1,6 +1,6 @@
 \part{Diffusion Processes}
 \label{part:diffusion}
 
-
-
 \input{./diffusion/index.tex}
+\input{./calculus/index.tex}
+\input{./sde/index.tex}

src/sde/exact.tex

@@ -0,0 +1,149 @@
+\section{Itô Existence and Uniqueness}
+\label{section:exact}
+
+\begin{definition}[Lipschitz Coefficient]
+\label{definition:lipschitz-coefficient}
+    Let $\sigma: [0, \infty) \times C([0, \infty); \real^d) \to \real^n$, then $\sigma$ is \textbf{Lipschitz} if there exists $C \ge 0$ such that for any $\theta, \eta \in C([0, \infty); \real^d)$, 
+    \[
+    \norm{\sigma(t, \theta) - \sigma(t, \eta)}_{\real^n} \le C\norm{\theta - \eta}_{u, [0, t]}
+    \]
+\end{definition}
+
+\begin{lemma}
+\label{lemma:exact-uniform-bound}
+    Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion, $\sigma$ be a $\bracs{\mathcal{F}_t}$-previsible $L(\real^d; \real^n)$-valued process, $b$ be a $\bracs{\mathcal{F}_t}$-previsible $\real^n$-valued process, $\xi \in L^p(\Omega, \cf_0; \real^n)$, and
+    \[
+    X_t = \xi + \int_0^t \sigma_s dB_s + \int_0^t b_s ds
+    \]
+
+    then for any $T > 0$ and $p \ge 2$, there exists $C_{T, p, n} \ge 0$ such that for all $0 \le t \le T$, 
+    \[
+    \ev\braks{\norm{X}_{u, [0, t]}^p} \le C_{T, p, n}\braks{\ev(\norm{\xi}_{\real^n}^p) 
+    + \ev\paren{\int_0^t \norm{\sigma_s}_{\real^n}^p + \norm{b_s}_{\real^n}^p ds }}
+    \]
+\end{lemma}
+\begin{proof}[Proof, \cite[Theorem 11.5]{Rogers}. ]
+    Assume without loss of generality that $\xi = 0$, then
+    \[
+    \norm{X}_{u, [0, t]}^p \le C_p\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p + \paren{\int_0^t \norm{b_s}_{\real^n}ds}^p
+    \]
+
+    where by Jensen's inequality, 
+    \[
+    \paren{\int_0^t \norm{b_s}ds}^p \le C_{t, p} \int_0^t \norm{b_s}_{\real^n}^p ds 
+    \]
+
+    and by the BDG inequality and Jensen's inequality, 
+    \[
+    \ev\braks{\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p} \le C_p \ev\braks{\paren{\int_0^t \norm{\sigma_s}_{\real^d}^2 ds}^{p/2}} \le C_p \int_0^t \norm{\sigma_s}_{\real^d}^p ds
+    \]
+\end{proof}
+
+\begin{lemma}
+\label{lemma:lipschitz-picard}
+    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} with constant $K$.
+
+    Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion. For any $\bracs{\mathcal{F}_t}$-adapted process $X: \Omega \to C([0, \infty); \real^d)$ with continuous sample paths and $\xi \in L^p(\Omega, \cf_0; \real^n)$, let
+    \[
+    P(X, \xi)_t = \xi + \int_0^t \sigma(s, X)dB_s + \int_0^t b(s, X)ds
+    \]
+    
+    then for any $\bracs{\mathcal{F}_t}$-adapted process $Y: \Omega \to C([0, \infty); \real^d)$, $\eta \in L^p(\Omega, \cf_0; \real^n)$, $T > 0$, and $0 \le t \le T$, 
+    \begin{align*}
+    &\ev\braks{\norm{P(X, \xi) - P(Y, \eta)}_{u, [0, t]}^p} \\
+    &\le C_{K, n, T, p}\braks{\norm{\xi - \eta}_{L^p(\Omega; \real^n)}^p + \ev\paren{\int_0^t \norm{X - Y}_{u, [0, s]}^p ds}}
+    \end{align*}
+    
+\end{lemma}
+
+\begin{theorem}[Itô]
+\label{theorem:ito-existence-uniqueness}
+    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
+    \[
+    X_t = \xi + \int_0^t \sigma(s, X)dB_s + \int_0^t b(s, X)ds
+    \]
+
+    is exact and pathwise unique. 
+\end{theorem}
+\begin{proof}[Proof of existence, \cite[Theorem 11.2]{Rogers}. ]
+    For each $t \ge 0$, let $\mathcal{G}_t^\circ = \sigma(\bracs{B_s|1 \le s \le t} \cup \bracs{\xi})$, $\mathcal{G}^\circ = \sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, $\mathcal{N}$ be the collection of $\bp$-null sets in the completion of $\sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, and $\mathcal{G}_t = \sigma(\mathcal{G}_t^\circ \cup \mathcal{N})$. 
+
+    Let $\xi \in L^\infty(\Omega, \cf_0; \real^n)$ and $X^{(0)} = \xi$. For each $m \in \natz$, define inductively
+    \[
+    X^{(m+1)} = P(X^{(m)}, \xi)_t = \xi + \int_0^t \sigma(s, X^{(m)})dB_s + \int_0^t b(s, X^{(m)})ds
+    \]
+
+    then $X^{(m)}$ has continuous sample paths, and for each $T > 0$, $\ev[{\normn{X^{(m)}}_{u, [0, s]}^2}] < \infty$. By \autoref{lemma:lipschitz-picard}, 
+    \[
+    \ev\braks{\norm{X^{(m+2)} - X^{(m+1)}}_{u, [0, T]}^2} \le C \int_0^t \ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, t]}^2} dt
+    \]
+
+    Therefore
+    \[
+    \ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, T]}^2} \le C_0 C \frac{T^n}{n!}
+    \]
+
+    and $X^{(m)}$ converges uniformly in $L^2$ to a limiting process $X$, which satisfies the given equation. 
+\end{proof}
+\begin{proof}[Proof of uniqueness. ]
+    Let $X$ and $Y$ be solutions of the SDE with the same setup, then by \autoref{lemma:lipschitz-picard},
+    \[
+    \ev\braks{\norm{X - Y}_{u, [0, T]}^2} \le C \int_0^t \ev\braks{\norm{X - Y}_{u, [0, t]}^2} dt
+    \]
+
+    which implies that $X|_{[0, T]} = Y|_{[0, T]}$ almost surely. 
+\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

@@ -0,0 +1,8 @@
+\chapter{Stochastic Differential Equations}
+\label{chap:sde}
+
+\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

@@ -0,0 +1,128 @@
+\section{Definitions}
+\label{section:sde-definitions}
+
+\begin{definition}[Previsible $\sigma$-Algebra]
+\label{definition:previsible-sigma-algebra}
+    Let $(\Omega, \cf)$ be a measurable space and $\bracs{\cf_t|t \ge 0}$ be a filtration on $\Omega$, then the \textbf{previsible $\sigma$-algebra} $\mathscr{J}_\omega$ on $(0, \infty) \times \Omega$ associated with $\bracs{\mathcal{F}_t}$ is the $\sigma$-algebra generated by
+    \[
+    \bracs{(s, t] \times A| 0 \le s < t < \infty, A \in \cf_s}
+    \]
+
+    In other words, it is the smallest $\sigma$-algebra on $(0, \infty) \times \Omega$ on which every $\bracs{\mathcal{F}_t}$-adapted process with left-continuous sample paths is measurable. 
+\end{definition}
+
+\begin{definition}[Previsible Path Functional]
+\label{definition:previsible-path-functional}
+    Let $C([0, \infty); \real^d)$ be the space of $\real^d$-valued continuous functions on $[0, \infty)$, equipped with the topology of uniform convergence. For each $t \ge 0$, let
+    \[
+    \mathscr{X}_t = \sigma(\bracs{\pi_s|s \le t})
+    \]
+
+    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
+    \begin{enumerate}
+        \item $X$ is $(\mathscr{J}_\Omega, \mathscr{J}_{C([0, \infty); \real^d)})$-measurable. 
+        \item For any previsible path functional $\alpha: (0, \infty) \times C([0, \infty); \real^d)$, $\alpha(t, X)$ is $\bracs{\mathcal{F}_t}$-previsible. 
+    \end{enumerate}
+\end{lemma}
+
+\begin{definition}[Diffusion Type SDE]
+\label{definition:diffusion-sde}
+    Let $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then a \textbf{SDE of diffusion type} is the equation
+    \[
+    X_t = \xi + \int_0^t \sigma(X_s) dB_s + \int_0^t b(X_s)ds
+    \]
+
+    under the constraint
+    \begin{equation}
+    \int_0^t \norm{\sigma(X_s)}_{\real^n}^2 + \norm{b(X_s)}_{\real^n} ds < \infty \label{equation:diffusion-constraint}
+    \end{equation}
+
+    for all $t > 0$, where
+    \begin{itemize}
+        \item $B$ is a standard Brownian motion on a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$.
+        \item $\xi$ is a $\cf_0$-measurable random variable. 
+    \end{itemize}
+\end{definition}
+
+\begin{definition}[Pathwise Uniqueness]
+\label{definition:pathwise-uniqueness}
+    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(s, X) dB_s + \int_0^t b(s, X) ds\label{equation:diffusion-sde}
+    \end{equation}
+
+    has \textbf{pathwise uniqueness} if given
+    \begin{itemize}
+        \item A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,
+        \item $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,
+        \item Continuous $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^d)$ satisfying \autoref{equation:diffusion-sde} and \autoref{equation:diffusion-constraint},
+    \end{itemize}
+
+    then $X = Y$ almost surely. 
+\end{definition}
+
+\begin{definition}[Pathwise Exact]
+\label{definition:pathwise-exact}
+    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
+    \[
+    X_t = \xi + \int_0^t \sigma(s, X) dB_s + \int_0^t b(s, X) ds
+    \]
+
+    is \textbf{pathwise exact} if given
+    \begin{itemize}
+        \item A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,
+        \item $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,
+        \item $\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^d)$ satisfying \autoref{equation:diffusion-sde} and \autoref{equation:diffusion-constraint},
+    \end{itemize}
+
+    then for every $t \ge 0$, $X_t = Y_t$ almost surely. 
+\end{definition}
+
+
+
+
+\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}
+
+