Brownian motionBrownianMotion2005-04-27 16:48:142008-10-19 17:09:33DefinitionLevy characterization% this is the default PlanetMath preamble. as your knowledge
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\begin{mydefn}
One-dimensional \emph{Brownian motion} is a stochastic process $W(t)$, defined for $t\in [0,\infty)$ such that
\begin{enumerate}
\item $W(0) = 0$ almost surely
\item The sample paths $t \mapsto W(t)$ are almost surely continuous.
\item For any finite sequence of times $t_0 < t_1 < \cdots <t_n$,
the increments
\[
W(t_1)-W(t_0), W(t_2)-W(t_1), \dotsc, W(t_n)-W(t_{n-1})
\]
are independent.
\item
For any times $s < t$, $W(t)-W(s)$ is normally distributed with mean zero
and variance $t-s$.
\end{enumerate}
\end{mydefn}
\begin{mydefn}
A $d$-dimensional Brownian motion is a stochastic process
$W(t) = (W_1(t), \dots, W_d(t))$ in
$\mathbb{R}^d$ whose coordinate processes $W_i(t)$ are
independent one-dimensional Brownian motions.
\end{mydefn}
\begin{figure}
\includegraphics{brownian.eps}
\caption{Sample paths of a standard Brownian motion}
\end{figure}