Brownian Motion (Wiener Process)

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Brownian Motion (Wiener Process)</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="0%" align="left" valign="top"><p class="word_list"> <a href="#Louis Bachelier">Bachelier, Louis</a><p class="word_list"> <a href="#Brownian motion">Brownian motion</a><p class="word_list"> <a href="#Norbert Wiener">Wiener, Norbert</a><p class="word_list"> <a href="#Wiener process">Wiener process</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <p class="body"><b><a name="Brownian motion">Brownian motion</a></b> is a simple <a href="../articles/time_series_stochastic_process.htm">continuous stochastic process</a> that is widely used in physics and finance for modeling random behavior that evolves over time. Examples of such behavior are the random movements of a molecule of gas or fluctuations in an asset's price. </p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table5"> <tr> <td width="160"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> </td> <td width="11"> </td> </tr> </table> <p class="body">Intuitively, we may think of Brownian motion as a limiting case of some <a href="../articles/random_walk.htm">random walk</a> as its time increment goes to zero. This is illustrated in Exhibit 1.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="325" cellpadding="0" id="table6"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Brownian Motion as a Limiting Case of a Random Walk</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_brownian_motion_a.gif" width="300" height="205"><br> <img border="0" src="../images/ex1_brownian_motion_b.gif" width="300" height="205"></td> </tr> <tr> <td> <p class="exhibit_legend">Intuitively, we may think of a Brownian motion as a limiting case of some random walk as its time increment goes to zero. The upper graph depicts a realization of a random walk. The lower graph depicts a similar realization of a Brownian motion.</td> </tr> </table> </center> </div> <p class="body">Let's formalize this. If you have not already done so, see the <i> <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation conventions</a></i> documentation. A univariate Brownian motion is defined as a stochastic process <i><b>B</b></i> satisfying</p> <p class="bullet_line_space">1. The process is defined for times <i>t</i> <img border="0" src="../symbols/greater_than_or_equal.gif" align="absbottom" width="9" height="14"> 0, with <sup>0</sup><i>B</i> = 0.</p> <p class="bullet_line_space">2. Realizations are continuous functions of time <i>t</i>.</p> <p class="bullet_line_space">3. Random variables <i><sup>t</sup>B</i> – <i> <sup>s</sup>B</i> are normally distributed with mean 0 and variance <i>t</i> – <i>s</i>, for <i>t</i> > <i>s</i>.</p> <p class="bullet_line_space">4. Random variables <i><sup>t</sup>B</i> – <i> <sup>s</sup>B</i> and <i><sup>v</sup>B</i> – <i><sup>u</sup>B</i> are independent whenever <i>v</i> > <i>u</i> <img border="0" src="../symbols/greater_than_or_equal.gif" align="absbottom" width="9" height="14"> <i>t</i> > <i>s</i> <img border="0" src="../symbols/greater_than_or_equal.gif" align="absbottom" width="9" height="14"> 0.</p> <p class="body">Brownian motion is a <a href="../articles/martingale.htm">martingale</a>. It has a number of other interesting properties. One is that realizations, while continuous, are differentiable nowhere with probability 1. Realizations are fractals. No matter how much you magnify a portion of a realization, the result still looks like a realization of a Brownian motion. This is illustrated in Exhibit 2.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="425" cellpadding="0" id="table7"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Realizations of a Brownian motion are fractals</span><br> <span class="exhibit_subheader">Exhibit 2</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td align="left"> <img border="0" src="../images/ex2_brownian_motion.gif" width="392" height="215"></td> </tr> <tr> <td> <p class="exhibit_legend">Realizations of Brownian motions have the same jagged appearance no matter how much you magnify them.</td> </tr> </table> </center> </div> <p class="body">Brownian motion can easily be generalized to multiple dimensions. An <i>n</i>-dimensional Brownian motion is simply an <i>n</i>-dimensional vector of <i>n</i> independent Brownian motions.</p> <p class="body">Brownian motion gets its name from the botanist Robert Brown (<a href="#Brown">1828</a>) who observed in 1827 how particles of pollen suspended in water moved erratically on a microscopic scale—first moving in one direction and then zig zagging in another. The motion was caused by water molecules randomly buffeting the particle of pollen. Brown posed the problem of mathematically describing the observed movement, but he did not solve the problem himself.</p> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="left" id="table8"> <tr> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="9"> </td> </tr> </table><p class="body">The first discoverer of the stochastic process that we today call Brownian motion was <b><a name="Louis Bachelier">Louis Bachelier</a></b>. Anticipating by 70 years developments in <a href="../articles/option_pricing_theory.htm">options pricing theory</a>, Bachelier mathematically defined Brownian motion and proposed it as a model for asset price movements. He published these ideas in his (<a href="#Bachelier">1900</a>) doctoral thesis on speculation in the French <a href="../articles/coupon_bond.htm">bond</a> market. That work attracted little attention. Five years later, Albert Einstein (<a href="#Einstein">1905</a>) independently discovered the same stochastic process and applied it in thermodynamics. The work of Bachelier and Einstein was not entirely rigorous. Neither man proved that a stochastic process even existed satisfying the four properties that define Brownian motion. <b><a name="Norbert Wiener">Norbert Wiener</a></b> (<a href="#Wiener">1923</a>) ultimately proved the existence of Brownian motion and made significant contributions to related mathematical theories, so Brownian motion is often called a <b> <a name="Wiener process">Wiener process</a></b>.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table9"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Internal Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table10"> <tr> <td width="100%"> <span style="font-family: Times New Roman; letter-spacing: -0.2pt"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/joint_normal_distribution.htm">joint normal distribution</a> A multivariate distribution with normal marginal distributions.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/martingale.htm">martingale</a> A type of stochastic process that has zero drift.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/normal_distribution.htm"><strong style="font-weight: 400">normal distribution</strong></a><strong style="font-weight: 400"> Perhaps the most important probability distribution for probability and statistics.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/option_pricing_theory.htm">option pricing theory</a> An introductory article.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/random_walk.htm">random walk</a> A discrete stochastic process whose increments form a white noise.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/time_series_stochastic_process.htm">time series and stochastic processes</a> An introductory article.</strong></p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/volatility.htm">volatility</a> A metric of  variability in a stochastic process.</strong></p> </span> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_3.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Courses</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table11"> <tr> <td width="100%"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table12"> <tr> <td width="127" bgcolor="#669933" align="left" valign="top"> <img border="0" src="../images/picture_course_math.jpg" width="125" height="125"></td> <td width="100%" align="left" valign="top"> <p class="explanatory_text"><b> <font size="2"> <a target="_blank" href="http://www.risklearning.com/link/math_3.htm"> Financial Math 3</a></font></b></p> <p class="explanatory_text">This is the third in a series of three-day courses on financial mathematics that take participants from pre-calculus to stochastic calculus. Math 3 covers statistics, time series, stochastic calculus, and plenty of financial applications. Math 3 is a fun, engaging, and enlightening look at the fascinating field of financial math. </td> </tr> </table> </div> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Cited Papers</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table13"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Bachelier">Bachelier</a>, Louis (1964). Theory of Speculation, <i>The Random Character of Stock Prices</i>, Paul H. Cootner (editor), Cambridge: MIT. Translation of Bachelier's 1900 doctoral thesis.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Brown">Brown</a>, Robert (1828). A Brief account of microscopical observations made in the months of June, July and August 1827 on the particles contained in the pollen of plants, <i>privately circulated</i>.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Einstein">Einstein</a>, Albert (1905). Uber die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, <i>Annalen der Physik und Chemie</i>, 17 (4), 549-560 </p> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Wiener">Wiener</a>, Norbert (1923), Differential space, <i>Journal of Mathematical Physics</i> 2, 131-174 </p> </td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_1.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%">  <p class="ad_legend" style="margin-left: 0; margin-bottom:2"> <a target="_top" href="http://www.riskexpertise.com/link/advertising.htm"> <font color="#000000">Ads by Contingency Analysis</font></a></p> <p style="margin-top: 0; margin-bottom: 0"> <iframe src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=iframe" style="margin: 0px; padding: 0px; border-width: 0px" width="336" height="265" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" name="I1"> <![if lt IE 4]> <script src="http://servedbyadbutler.com/adserve/;ID=147917;size=336x265;setID=82436;type=js" type="text/javascript"> </script> <noscript> <a href="http://servedbyadbutler.com/go2/;ID=147917;size=336x265;setID=82436"> <img src="http://servedbyadbutler.com/adserve.ibs/;ID=147917;size=336x265;setID=82436;type=img" border="0" height="265" width="336" alt=""></a> </noscript><![endif]></iframe>   </td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table14"> <tr> <td width="100%" bgcolor="#FFFFFF"> <p align="center" style="margin-top: 18"><map name="FPMap365"> <area target="_top" href="http://www.riskglossary.com" shape="rect" coords="0, 0, 164, 102"> </map><img border="0" src="../images/logo.jpg" usemap="#FPMap365" width="165" height="103"></p> <p align="center"><span class="verdana10"> <a href="http://www.riskglossary.com/main/disclaimer.htm" target="_self"> Disclaimer</a></span></p> <p align="center">website: <span class="verdana10"> <a target="_top" href="http://www.contingencyanalysis.com"> http://www.contingencyanalysis.com</a></span><br> glossary direct link: <span class="verdana10"> <a target="_top" href="http://www.riskglossary.com"> http://www.riskglossary.com</a></span><br> copyright © Contingency Analysis, 2006<p align="center"><map name="FPMap1"> <area target="_top" href="http://www.contingencyanalysis.com" shape="rect" coords="0, 0, 240, 88"> </map><img border="0" src="../images/logo_glossary_white.gif" usemap="#FPMap1" width="241" height="89"></td> </tr> </table> </td> </tr> </table> </td> <td>  </td> </tr> </table> </div> </body>