Brownian motion: definition and properties
I choose to open this series of notes with a discussion of Brownian motion, arguably the most popular stochastic process in the theory of SDE. Prerequisites for this part are a basic knowledge of probability and measure theory.
Stochastic processes
A stochastic process is a family of random variables
\[ X=\{X_t\}_{t\in T} \]
defined on the same probability space
\[ (\Omega,\mathcal{F},\mathbb{P}), \]
with values in some state space \(E\).
Usually,
\[ T=[0,\infty) \]
and
\[ E=\mathbb{R}^d. \]
The process can equivalently be viewed as a function of two variables,
\[ X:T\times\Omega\to E, \]
with
\[ X(t,\omega)=X_t(\omega). \]
There are two complementary ways to look at it.
For fixed \(t\),
\[ \omega\mapsto X_t(\omega) \]
is a random variable.
For fixed \(\omega\),
\[ t\mapsto X_t(\omega) \]
is a function of time, called a sample path, trajectory, or realization of the process.
Stationary processes
A process is called stationary when its statistical behaviour does not change under shifts in time.
There are two important notions.
Strict stationarity
A stochastic process \(\{X_t\}\) is strictly stationary if, for every \(n\), every choice of times \(t_1,\ldots,t_n\), and every time shift \(h\),
\[ (X_{t_1},\ldots,X_{t_n}) \overset{d}{=} (X_{t_1+h},\ldots,X_{t_n+h}), \]
whenever the shifted times remain in the time domain.
Thus all finite-dimensional distributions are invariant under time translation.
Strict stationarity is therefore a statement about the entire law of the process.
Weak stationarity
Suppose \(X_t\in L^2\).
The process is weakly stationary, or second-order stationary, if
\[ \mathbb{E}[X_t] = \mu \]
is independent of \(t\), and the covariance depends only on the time difference:
\[ \operatorname{Cov}(X_t,X_s) = C(t-s). \]
For a centered process,
\[ \mathbb{E}[X_t]=0, \]
this becomes
\[ C(t-s) = \mathbb{E}[X_tX_s]. \]
The function \(C\) is called the autocovariance or correlation function.
Thus weak stationarity says that the first two moments are invariant under time translation.
Strict stationarity is stronger in general:
\[ \text{strict stationarity} + X_t\in L^2 \quad\Longrightarrow\quad \text{weak stationarity}. \]
The converse is generally false.
Independent increments
\(X_t\) has independent increments if, for all
\[ t_0 < t_1 < \cdots < t_n, \]
the random variables
\[ X_{t_1}-X_{t_0}, \quad X_{t_2}-X_{t_1}, \quad \ldots \]
are independent.
If, for all \(t_1,t_2,s\in T\) and \(B\subset\mathbb R\),
\[ \mathbb P(X_{t_1+s}-X_{t_1}\in B) = \mathbb P(X_{t_2+s}-X_{t_2}\in B), \]
then \(X_t\) has stationary independent increments.
Brownian motion
A Brownian motion
\[ W(t):\mathbb R^+\to\mathbb R \]
is a real-valued stochastic process with a.s. continuous paths such that
\[ W(0)=0, \]
it has independent increments, and, for all \(t>s\geq 0\),
\[ W(t)-W(s) \]
has a Gaussian distribution with mean \(0\) and variance \(t-s\):
\[ W(t)-W(s)\sim\mathcal N(0,t-s). \]
The density of the random variable \(W(t)-W(s)\) is
\[ g(x;t,s) = \frac{1}{\sqrt{2\pi(t-s)}} \exp\left( -\frac{x^2}{2(t-s)} \right). \]
Although Brownian motion has stationary increments, it is not a stationary process. The probability density of the one-dimensional Brownian motion is
\[ g(x,t) = \frac{1}{\sqrt{2\pi t}} \exp\left( -\frac{x^2}{2t} \right). \]
Similarly, a \(d\)-dimensional Brownian motion
\[ W(t):\mathbb R^+\to\mathbb R^d \]
is a vector of \(d\) independent Brownian motions,
\[ W(t) = \left( W_1(t),\ldots,W_d(t) \right). \]
The density of the Gaussian random variable \(W(t)-W(s)\) is
\[ g(x;t,s) = \frac{1}{[2\pi(t-s)]^{d/2}} \exp\left( -\frac{\|x\|^2}{2(t-s)} \right). \]
For \(d\)-dimensional Brownian motion,
\[ \mathbb E[W(t)] = 0, \qquad \forall t\geq 0, \]
and
\[ \mathbb E \left[ (W(t)-W(s)) \otimes (W(t)-W(s)) \right] = (t-s)I. \]
Moreover,
\[ \mathbb E \left[ W(t)\otimes W(s) \right] = \min(t,s)I. \]
Properties of Brownian motion
The following properties will be central to the development of stochastic calculus. I choose to present them in a non-technical way, using heuristic arguments rather than proofs.
Hölder continuity
Brownian paths are almost surely locally Hölder continuous with any exponent
\[ \alpha \in \left(0,\frac12\right). \]
That is, for every \(T>0\) and \(\alpha<1/2\), there exists an a.s. finite random constant \(C\) such that
\[ |W_t-W_s| \leq C|t-s|^\alpha, \qquad s,t\in[0,T]. \]
Brownian motion has unbounded variation
For a partition
\[ \Pi=\{0=t_0<t_1<\cdots<t_n=t\}, \]
define
\[ \Delta W_k=W_{t_{k+1}}-W_{t_k}, \qquad \Delta t_k=t_{k+1}-t_k, \]
Since (remember that \(\Delta W_k\sim\mathcal N(0,\Delta t_k)\))
\[ \mathbb E[(\Delta W_k)^2]=\Delta t_k, \qquad \mathbb E[(\Delta W_k)^4]=3(\Delta t_k)^2, \]
we have
\[ \mathbb E[\sum_{t_k<t}(\Delta W_k)^2] = \sum_{t_k<t}\Delta t_k = t. \]
Moreover, by independence of the increments,
\[ \begin{aligned} \mathbb E[(\sum_{t_k<t}(\Delta W_k)^2-t)^2] &= \sum_{t_k<t} \mathbb E \left[ \left((\Delta W_k)^2-\Delta t_k\right)^2 \right] \\ &= \sum_{t_k<t} \left( 3(\Delta t_k)^2-(\Delta t_k)^2 \right) \\ &= 2\sum_{t_k<t}(\Delta t_k)^2 \longrightarrow 0. \end{aligned} \]
Hence
\[ \sum_{t_k<t}(\Delta W_k)^2. = \sum_{t_k<t}(\Delta W_k)^2 \longrightarrow t \qquad\text{in }L^2. \]
Now suppose, by contradiction, that a Brownian path has bounded variation on \([0,t]\). Then
\[ V_t(W) = \sup_{\Pi} \sum_{t_k<t}|\Delta W_k| <\infty, \]
so, for every partition \(\Pi\),
\[ \sum_{t_k<t}|\Delta W_k| \leq V_t(W). \]
Moreover,
\[ \begin{aligned} \sum_{t_k<t}(\Delta W_k)^2 &= \sum_{t_k<t}|\Delta W_k|\,|\Delta W_k| \\ &\leq \max_{t_k<t}|\Delta W_k| \sum_{t_k<t}|\Delta W_k| \\ &\leq \max_{t_k<t}|\Delta W_k|\,V_t(W). \end{aligned} \]
As \(|\Pi|\to0\), continuity of the Brownian path on the compact interval \([0,t]\) implies uniform continuity, and therefore
\[ \max_{t_k<t}|\Delta W_k| \longrightarrow0. \]
Since \(V_t(W)<\infty\) by assumption,
\[ 0 \leq \sum_{t_k<t}(\Delta W_k)^2 \leq \max_{t_k<t}|\Delta W_k|\,V_t(W) \longrightarrow0. \]
Hence a continuous path of bounded variation must have zero quadratic variation:
\[ \sum_{t_k<t}(\Delta W_k)^2\longrightarrow0. \]
But for Brownian motion we have just shown that
\[ \sum_{t_k<t}(\Delta W_k)^2\longrightarrow t. \]
For \(t>0\) this is a contradiction. Therefore
\[ V_t(W) = \sup_{\Pi} \sum_{t_k<t}|W_{t_{k+1}}-W_{t_k}| = \infty \qquad\text{a.s.} \]
Non-differentiability
Brownian paths are almost surely nowhere differentiable.
This should not be surprising: Brownian motion is continuous, but its paths have infinite variation on every non-trivial time interval.
Quadratic variation
We have just shown that, in \(L^2\),
\[ \sum_{t_k<t}(\Delta W_k)^2 \longrightarrow t. \]
This is conventionally written
\[ (dW_t)^2=dt \]
or, informally over a small increment,
\[ (\Delta W)^2\sim\Delta t. \]
Since the second-order term contributes at the same order as an ordinary time increment it cannot be discarded when Taylor-expanding functions of Brownian motion.
Two other increment rules
For a Brownian increment,
\[ \Delta W = W_{t+\Delta t}-W_t \sim \mathcal N(0,\Delta t). \]
Equivalently,
\[ \Delta W=\sqrt{\Delta t}\,Z, \qquad Z\sim\mathcal N(0,1). \]
Thus a Brownian increment is of order \(\sqrt{\Delta t}\). Consequently,
\[ (\Delta W)^2 = \Delta t\,Z^2, \]
is of order \(\Delta t\), while
\[ \Delta W\,\Delta t = (\Delta t)^{3/2}Z \]
and
\[ (\Delta t)^2 \]
are of higher order.
As before we could show that, in \(L^2\),
\[ \sum_k\Delta W_k\,\Delta t_k \longrightarrow 0, \qquad \sum_k(\Delta t_k)^2 \longrightarrow 0. \]
which gives
\[ dW_t\,dt=0, \qquad (dt)^2=0. \]
Brownian scaling
Let \(W_t\) be a one-dimensional Brownian motion and let \(c>0\). Define
\[ \frac{1}{c}W_{c^2t}. \]
is again a Brownian motion.
Equivalently, setting \(a=c^2\),
\[ \frac{1}{\sqrt a}W_{at} \overset{d}{=}W_t. \]
This is the Brownian scaling property.
Shift invariance
For every \(c>0\), the process
\[ X_t=W_{c+t}-W_c \]
is a Brownian motion independent of
\[ \{W_u:u\in[0,c]\}. \]
Time reversal
Let
\[ X_t=W_{1-t}-W_1, \qquad t\in[0,1]. \]
Then
\[ X_t\overset{d}{=}W_t. \]
Time inversion
Define
\[ X_0=0, \qquad X_t=tW_{1/t}, \quad t>0. \]
Then
\[ X_t\overset{d}{=}W_t. \]