\section{Brownian Motion}

\subsection{Brownian Motion}

\begin{definition}[standard Brownian motion]
A real-valued process $W = (W_t)_{t \in [0,T]}$, with $W_0 = 0$, defined on a filtered probability space $(\Omega, \F, \P)$, is called a standard $\F$-Brownian motion if:
\begin{enumerate}
    \item $W$ is $\F$-adapted.
    \item For any $0 \leq u \leq t \leq T$, $W_t - W_u$ is independent of $\F_u$.
    \item For any $0 \leq u \leq t \leq T$, $W_t - W_u \sim \N(0, t - u)$.
    \item $W$ is sample-paths continuous, i.e.\ almost all sample paths of $W$ are continuous.
\end{enumerate}
\end{definition}

\begin{definition}[Gaussian process]
A stochastic process $(X_t)_{t \geq 0}$ is said to be a Gaussian process, if for arbitrary times $0 < t_1 < t_2 < \cdots < t_n$, the vector $(X_{t_1}, \ldots, X_{t_n})$ follows a multivariate Gaussian distribution.
\end{definition}

\subsection{Properties of Brownian Motion}

\begin{theorem}[properties of Brownian motion]
Let $W$ be an $\F$-Brownian motion. Then
\begin{enumerate}
    \item \textbf{(martingale property)} $W$ is a continuous $\F$-martingale.
    \item \textbf{(Markov property)} For any bounded Borel measurable function $g : \R \to \R$ and $0 \leq u \leq t \leq T$,
    \[
    \E[g(W_t)|\F_u] = \E[g(W_t)|\sigma(W_u)] = \underbrace{h(W_u)}_{\text{Doob's measurability theorem}}
    \]
    \item $W$ is a Gaussian process.
    \item $\Cov(W_s, W_t) = \E[W_s W_t] = \min(s, t)$ for all $s, t \geq 0$ and in particular $\E[W_t^2] = t$.
\end{enumerate}
\end{theorem}

\begin{proof}
For (1), note that $W$ is an integrable, $\F$-adapted process with $\E(W_t|\F_u) = \E(W_t - W_u + W_u|\F_u) = W_u$ for any $0 \leq u \leq t \leq T$.

For (2),
\begin{align*}
\E[g(W_t)|\F_u] &= \E[g(W_t - W_u + W_u)|\F_u] \\
&= \E[g(W_t - W_u + x)|\F_u]\big|_{x=W_u} && \text{($W_u$ is $\F_u$-measurable)} \\
&= \E[g(W_t - W_u + x)]\big|_{x=W_u} && \text{(independent increments)}
\end{align*}
which is a function of $W_u$ and therefore $\sigma(W_u)$-measurable (Lemma~1.5.1). It follows that
\[
\E[\hblue{\E[g(W_t)|\F_u]}|\sigma(W_u)] = \hblue{\E[g(W_t)|\F_u]}
\]
\[
\Rightarrow \E[g(W_t)|\sigma(W_u)] = \E[g(W_t)|\F_u].
\]

For (3), note that
\[
\begin{pmatrix} W_{t_1} \\ W_{t_2} \\ \vdots \\ W_{t_n} \end{pmatrix}
=
\begin{pmatrix} 1 & 0 & \cdots & 0 \\ 1 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{pmatrix}
\underbrace{\begin{pmatrix} W_{t_1} \\ W_{t_2} - W_{t_1} \\ \vdots \\ W_{t_n} - W_{t_{n-1}} \end{pmatrix}}_{=: y \sim \N}
\]
We can then conclude by using the fact that any linear transform of a Gaussian vector is again Gaussian. In particular, $(W_{t_1}, \ldots, W_{t_n}) \sim \N(0, A\Sigma A^T)$ where $\Sigma = \Var(y)$.

For (4),
\begin{align*}
\Cov(W_s, W_t) &= \E[W_s W_t] \\
&= \E[W_s(W_t - W_s)] + \E[W_s^2] \\
&= \E[W_s]\E[W_t - W_s] + \E[W_s^2] \\
&= s
\end{align*}
Similarly when $s \geq t$, $\Cov(W_s, W_t) = t$. Therefore $\E[W_t W_s] = \min(s, t)$.
\end{proof}

\begin{examples}[Exponential Brownian motion with drift]
Let $x, m \in \R$ and $\sigma > 0$ and we set $Y_t = x e^{mt + \sigma W_t}$. The process $Y$ is called an exponential Brownian motion with drift $m$ and variance $\sigma^2$. We first notice that the process $Y$ is adapted to the filtration $\F$. Next,
\[
\E[Y_t] = x e^{mt} \E(e^{\sigma W_t}) = x e^{mt} M_{W_t}(\sigma) = x e^{(m + \sigma^2/2)t}.
\]
We want to show that the process $Y$ is an $\F$-martingale if and only if $m = -\sigma^2/2$. Indeed, for $s \leq t$,
\begin{align*}
\E[x e^{mt + \sigma W_t} | \F_s] &= \E[x e^{ms + m(t-s) + \sigma W_s + \sigma(W_t - W_s)} | \F_s] \\
&= Y_s \E[e^{m(t-s) + \sigma(W_t - W_s)}] \\
&= Y_s e^{m(t-s) + \sigma^2(t-s)/2}
\end{align*}
and the claim follows.
\end{examples}

\subsection{Quadratic Variation}

\subsubsection{Total Variation}

\begin{definition}[total variation, aka first-order variation]
Let $\pi_n = \{t_0 = 0, t_1, \ldots, t_{m_n} = T\}$ be a partition of $[0,T]$ with $n(\pi) = m_n$ subintervals. The maximum step size of the partition is denoted by
\[
|\pi_n| := \max_{1 \leq k \leq m_n} (t_k - t_{k-1}).
\]
For a given partition $\pi_n$, we define the corresponding measure of the oscillation of a function $f$ by
\[
V_n(f) = \sum_{k=1}^{m_n} |f(t_k) - f(t_{k-1})|.
\]
The total variation of $f$ is defined as
\[
V(f) := \lim_{|\pi_n| \to 0} V_n(f) = \lim_{|\pi_n| \to 0} \sum_{k=1}^{m_n} |f(t_k) - f(t_{k-1})|
\]
\end{definition}

\begin{lemma}
Any continuously differentiable function $f$ is a function of finite variation, i.e.\ $V(f) < \infty$.
\end{lemma}

\begin{proof}
\[
V_n(f) = \sum_{k=1}^{m_n} \left|\int_{t_{k-1}}^{t_k} f'(t) \, dt\right| \leq \sum_{k=1}^{m_n} \int_{t_{k-1}}^{t_k} |f'(t)| \, dt = \int_0^T |f'(t)| \, dt < \infty
\]
where we have used the fact that any continuous function on a closed and bounded interval is bounded. By taking the limit as $|\pi_n| \to 0$, we have proved that any continuously differentiable function $f$ on $[0,T]$ is a function of finite variation.
\end{proof}

\begin{theorem}[Jordan's decomposition and related results]
\leavevmode
\begin{itemize}
    \item Any bounded increasing function $f$ is a function of finite variation, i.e.\ $V(f) < \infty$.
    \item (Jordan's decomposition) If $f$ is a function of finite variation, i.e.\ $V(f) < \infty$, then $f = f_1 - f_2$ where $f_1, f_2$ are non-decreasing functions.
    \item The difference of two bounded increasing functions is a function of finite variation.
\end{itemize}
\end{theorem}

\begin{proof}
(Sketch) The first result comes from $V(f) = |f(T) - f(0)| < \infty$.

The second result can be proved by construction (omitted).

The third result follows from (1) as well as the fact that the difference of two functions of finite variation also has finite variation (a direct result of triangle inequality).
\end{proof}

\subsubsection{Quadratic Variation}

\begin{definition}[quadratic variation]
For a given partition $\pi_n = \{t_0 = 0, t_1, \ldots, t_{m_n} = T\}$ of $[0,T]$, the quadratic variation of $f$ is defined as
\[
\langle f \rangle = V^{(2)}(f) := \lim_{|\pi_n| \to 0} V_n^{(2)}(f) = \lim_{|\pi_n| \to 0} \sum_{k=1}^{m_n} |f(t_k) - f(t_{k-1})|^2
\]
\end{definition}

\begin{lemma}
Any continuously differentiable function $f$ has zero quadratic variation, i.e.\ $V^{(2)}(f) = 0$.
\end{lemma}

\begin{proof}
\[
V_n^{(2)}(f) = \sum_{k=1}^{m_n} |f(t_k) - f(t_{k-1})|^2 = \sum_{k=1}^{m_n} \left(\int_{t_{k-1}}^{t_k} f'(t) \, dt\right)^2
\]
\[
\leq \sum_{k=1}^{m_n} (t_k - t_{k-1}) \int_{t_{k-1}}^{t_k} |f'(t)|^2 \, dt \qquad \text{(by Cauchy--Schwarz, see Theorem~A.0.1)}
\]
\[
\leq |\pi_n| \int_0^T |f'(t)|^2 \, dt < \infty
\]
Taking the limit as $|\pi_n| \to 0$, we have $V^{(2)}(f) = \lim_{|\pi_n| \to 0} V_n^{(2)}(f) = 0$, completing the proof. Alternatively, this can also be shown by combining the results of Lemma~3.3.1 and~3.3.3.
\end{proof}

\begin{lemma}[finite variation $\Rightarrow$ zero quadratic variation]
Let $f$ be a continuous function of finite variation on $[0,T]$. Then the quadratic variation of $f$ is equal to zero.
\end{lemma}

\begin{proof}
Observe that for any partition $\pi_n = \{t_0 = 0, t_1, \ldots, t_{m_n} = T\}$ of $[0,T]$ we have
\begin{align*}
V_n^{(2)}(f) &= \sum_{k=1}^{m_n} |f(t_k) - f(t_{k-1})|^2 \\
&\leq \max_{1 \leq m \leq m_n} |f(t_m) - f(t_{m-1})| \sum_{k=1}^{m_n} |f(t_k) - f(t_{k-1})| \\
&= \max_{1 \leq m \leq m_n} |f(t_m) - f(t_{m-1})| \cdot V_n(f).
\end{align*}
By using the fact that any continuous function is uniformly continuous on a closed bounded interval, taking the limit as $|\pi_n| \to 0$ completes the proof.
\end{proof}

\begin{corollary}
(from Lemma~3.3.3) If the quadratic variation of a continuous stochastic process is a.s.\ strictly positive, then the total variation of the process is almost surely infinity. In particular, sample paths of a Brownian motion are a.s.\ functions of infinite variation on any interval.
\end{corollary}

\begin{theorem}
Let $W$ be a standard Brownian motion. $W$ is a process of finite quadratic variation. In particular, $\langle W \rangle_t = t$ (a.s.) for all $t \geq 0$.
\end{theorem}

\begin{proof}
Let $t \geq 0$ and $\pi_n = \{t_0 = 0, t_1, \ldots, t_{n(\pi)} = t\}$ be a partition of $[0,t]$. We want to show that
\[
\lim_{n(\pi) \to \infty} \sum_{k=1}^{n(\pi)} (W_{t_k} - W_{t_{k-1}})^2 =: V_n^{(2)}(W) = t
\]
where the convergence is in $L^2(\Omega, \F_T, \P)$, that is,
\[
\lim_{n(\pi) \to \infty} \E\left[\left(\sum_{k=1}^{n(\pi)} (W_{t_k} - W_{t_{k-1}})^2 - t\right)^2\right] = 0.
\]
For brevity of notation, write $\Delta t_k = t_k - t_{k-1}$, $\Delta W_k = W_{t_k} - W_{t_{k-1}}$ and $\theta_k = (\Delta W_k)^2 - \Delta t_k$. Our goal is to prove that $I_n = \E[(\sum \theta_k)^2] \to 0$ as $n(\pi) \to \infty$. To show this,
\[
I_n = \E\left[\left(\sum_{k=1}^{n(\pi)} \theta_k\right)^2\right] = \E\left[\sum_{i,j} \theta_i \theta_j\right] = \sum_{i=1}^{n(\pi)} \E(\theta_i^2) + 2\sum_{i \neq j} \E(\theta_i \theta_j)
\]
We can show that the second term is $0$ by using independent increments and $\E[(\Delta W_k)^2] = \Delta t_k$. Then
\begin{align*}
I_n &= \sum_{k=1}^{n(\pi)} \left(\E[(\Delta W_k)^4] - 2\Delta t_k \E[(\Delta W_k)^2] + (\Delta t_k)^2\right) \\
&= 2\sum_{k=1}^{n(\pi)} (\Delta t_k)^2 \leq 2|\pi_n| \sum_{k=1}^{n(\pi)} \Delta t_k = 2t|\pi_n|
\end{align*}
where in the second equality we used the fact the 4th central moment of a $X \sim \N(\mu, \sigma^2)$ r.v.\ is $\E((X - \mu)^4) = 3\sigma^4$.

It is now clear that the quantity $I_n$ tends to zero as $n$ tends to infinity, since $|\pi_n|$ tends to zero.
\end{proof}

\begin{lemma}
The process $M = W^2 - \langle W \rangle$ is a continuous $\F$-martingale.
\end{lemma}

\begin{proof}
The process $M$ is $\F$-adapted and integrable since $W^2$ is $\F$-adapted and integrable, while the quadratic variation $\langle W \rangle_t = t$ is clearly $\F$-adapted and integrable.

For $0 \leq u \leq t \leq T$, from the equality $\E(W_t W_u|\F_u) = W_u \E(W_t|\F_u) = W_u^2$, we deduce that
\begin{align*}
\E(M_t|\F_u) &= \E(W_t^2 - t|\F_u) \\
&= \E(W_t^2 - W_u^2 + W_u^2 - t|\F_u) \\
&= \E(W_t^2 - W_u^2|\F_u) + W_u^2 - t \\
&= \E((W_t - W_u)^2|\F_u) + W_u^2 - t \\
&= W_u^2 - u = M_u.
\end{align*}
\end{proof}

\begin{theorem}[alternative definition of quadratic variation for martingales]
Let $M$ be a continuous, square integrable $\F$-martingale, i.e.\ $\E(M_t^2) < \infty$ for all $t \geq 0$. Then the quadratic variation $\langle M \rangle$ is the unique continuous, increasing and $\F$-adapted process with $\langle M \rangle_0 = 0$ such that $M^2 - \langle M \rangle$ give us an $\F$-martingale.
\end{theorem}

\begin{remark}
The existence of $\langle M \rangle$ in the theorem above follows from the Doob--Meyer decomposition (Theorem~2.4.2).
\end{remark}

We now obtain a generalisation of Lemma~3.3.4:

\begin{corollary}
(from Theorem~3.3.3) If $M$ is a continuous, square integrable $\F$-martingale, then
\[
M^2 - \langle M \rangle
\]
is also an $\F$-martingale. In particular, if $M_0 = 0$, then $\E[M_t^2 - \langle M \rangle_t] = M_0^2 - \langle M \rangle_0 = 0$ and therefore
\[
\Var(M_t) = \E(M_t^2) = \E(\langle M \rangle_t).
\]
\end{corollary}

\subsection{Reflection Principle}

Let $b > 0$ and $B$ a standard Brownian motion. Define the hitting time
\[
T_b = \inf\{s : B_s = b\}.
\]

\begin{theorem}
The following results hold (pp.41--44):
\begin{itemize}
    \item \textbf{(reflection principle)} $\P(T_b < t) = 2\P(B_t > b)$. More precisely, $T_b$ follows a L\'{e}vy distribution with parameters $(0, b^2)$ (or equivalently, an inverse Gamma distribution with shape $1/2$ and scale $b^2$).
    \item We can therefore show that $\P(T_b < \infty) = 1$ but $\E(T_b) = \infty$.
    \item The moment generating function of $T_b$ is $e^{-\sqrt{2\lambda}\, b}$ for $b, \lambda > 0$.
    \item For any $t \geq 0$, the running supremum $B_t^* = \sup_{s \leq t} B_s$ and $|B_t|$ have the same distribution.
    \item The joint distribution of $B_t^* = \sup_{s \leq t} B_s$ and $B_t$ is given by
    \[
    \P(B_t \leq x, B_t^* > y) = \P(B_t > 2y - x) = \P(B_t < x - 2y)
    \]
    for every $t \geq 0$, $y \geq 0$ and $x \leq y$.
    \item The joint density of $B_t^* = \sup_{s \leq t} B_s$ and $B_t$ is given by
    \[
    \frac{2(2y - x)}{t} \cdot \frac{1}{\sqrt{2\pi t}} \exp\!\left(-\frac{(2y - x)^2}{2t}\right)
    \]
    for $x \leq y$, $y \geq 0$.
    \item The drawdown $B_t^* - B_t$ has the same distribution as $|B_t|$.
\end{itemize}
\end{theorem}