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10
.vscode/project.code-snippets vendored
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@@ -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)"]
}
}

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refs.bib
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@@ -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}
}

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src/calculus/frobenius.tex Normal file
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\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}

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\chapter{Integral Manifolds/Shenanigans}
\label{chap:integral}
\input{./integrable.tex}
\input{./frobenius.tex}

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src/calculus/integrable.tex Normal file
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@@ -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}

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src/diffusion/martingale.tex
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@@ -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}

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

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@@ -0,0 +1,103 @@
\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}

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\chapter{Stochastic Differential Equations}
\label{chap:sde}
\input{./setup}
\input{./exact}

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\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{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 $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then the diffusion 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}
\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 $\sigma: \real^d \to L(\real^d; \real^n)$ and $b: \real^n \to \real^n$ be measurable functions, then the diffusion 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}
\end{equation}
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}