\documentclass[border=10pt]{standalone}
%%%<
\usepackage{verbatim}
%%%>
\usepackage{pgfplots}
\pgfplotsset{width=7cm,compat=1.8}
\usepackage{pgfplotstable}
\begin{comment}
:Title: Brownian motions 
:Tags: 2D;Random;PGFPlotstable
:Author: Jake
:Slug: brownian-motions

Here's an approach using pgfplotstable to calculate the Brownian motions
as cumulative sums of random normally distributed values. You have to first
initialize an empty table, using something like
\pgfplotstablenew{200}\loadedtable, and then you can draw the brownian motions
using \addplot table [brownian motion] {\loadedtable};.

You can set the initial value and the maximum and minimum values using
\addplot table [brownian motion={start=0.5, min=-1, max=1}] {\loadedtable};

This code was written by Jake on TeX.SE.
\end{comment}
% Create a function for generating inverse normally distributed numbers using the Box–Muller transform
\pgfmathdeclarefunction{invgauss}{2}{%
  \pgfmathparse{sqrt(-2*ln(#1))*cos(deg(2*pi*#2))}%
}
% Code for brownian motion
\makeatletter
\pgfplotsset{
    table/.cd,
    brownian motion/.style={
        create on use/brown/.style={
            create col/expr accum={
                (\coordindex>0)*(
                    max(
                        min(
                            invgauss(rnd,rnd)*0.1+\pgfmathaccuma,
                            \pgfplots@brownian@max
                        ),
                        \pgfplots@brownian@min
                    )
                ) + (\coordindex<1)*\pgfplots@brownian@start
            }{\pgfplots@brownian@start}
        },
        y=brown, x expr={\coordindex},
        brownian motion/.cd,
        #1,
        /.cd
    },
    brownian motion/.cd,
            min/.store in=\pgfplots@brownian@min,
        min=-inf,
            max/.store in=\pgfplots@brownian@max,
            max=inf,
            start/.store in=\pgfplots@brownian@start,
        start=0
}
\makeatother

% Initialise an empty table with a certain number of rows
\pgfplotstablenew{201}\loadedtable % How many steps?
\pgfplotsset{
        no markers,
        xmin=0,
        enlarge x limits=false,
        scaled y ticks=false,
        ymin=-1, ymax=1
}
\tikzset{line join=bevel}
\begin{document}
\pgfmathsetseed{3}
\begin{tikzpicture}
  \begin{axis}
   \addplot table [brownian motion] {\loadedtable};
   \addplot table [brownian motion] {\loadedtable};
  \end{axis}
\end{tikzpicture}

\pgfmathsetseed{3}
\begin{tikzpicture}
  \begin{axis}
    \addplot table [
        brownian motion = {%
            max =  0.5,
            min = -0.75
        }
    ] {\loadedtable};
    \addplot table [
        brownian motion = {%
            start =  0.5,
            min   = -0.5,
            max   =  0.75
        }
    ] {\loadedtable};
  \end{axis}
\end{tikzpicture}
\end{document}