Brownian Motion

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Definition: Brownian Motion


Brownian Motion

Quick Summary of Brownian Motion


A theory or model that is used to explain random motion. Traditionally, Brownian motion was developed to explain the random movement seen in suspended particles, but is commonly applied today to the stock market. Brownian motion is just one theory that attempts to explain stock market fluctuations, along with the random walk theory and Markov process.




Full Definition of Brownian Motion


Brown­ian mo­tion is a sim­ple con­tin­u­ous sto­chas­tic process 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. Brown­ian mo­tion gets its name from the botanist Robert Brown (1828) who ob­served in 1827 how par­ti­cles of pollen sus­pended in water moved er­rat­i­cally on a mi­cro­scopic scale. 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.

In­tu­itively, we may think of Brown­ian mo­tion as a lim­it­ing case of some ran­dom walk as its time in­cre­ment goes to zero. This is il­lus­trated in Ex­hibit 1.

Ex­hibit 1: 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.

Let’s for­mal­ize this. A uni­vari­ate Brown­ian mo­tion is de­fined as a sto­chas­tic process B sat­is­fy­ing

  1. The process is de­fined for times t ≥ 0, with 0B = 0.
  2. Re­al­iza­tions are con­tin­u­ous func­tions of time t.
  3. Ran­dom vari­ables tB – sB are nor­mally dis­trib­uted with mean 0 and vari­ance t – s, for t > s.
  4. Ran­dom vari­ables tB – sB and vB – uB are in­de­pen­dent when­ever v > u ≥ t > s ≥ 0.

Brown­ian mo­tion is a mar­tin­gale. 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 graph of a re­al­iza­tion, the re­sult still looks like a re­al­iza­tion of a Brown­ian mo­tion.

Brown­ian mo­tion can eas­ily be gen­er­al­ized to mul­ti­ple di­men­sions. An n-di­men­sional Brown­ian mo­tion is sim­ply an n-di­men­sional vec­tor of n in­de­pen­dent Brown­ian mo­tions.

The first dis­cov­erer of the sto­chas­tic process that we today call Brown­ian mo­tion was Louis Bache­lier. An­tic­i­pat­ing by 70 years de­vel­op­ments in op­tions pric­ing the­ory, Bache­lier math­e­mat­i­cally de­fined Brown­ian mo­tion and pro­posed it as a model for asset price move­ments. He pub­lished these ideas in his (1900) doc­toral the­sis on spec­u­la­tion in the French bond mar­ket. That work at­tracted lit­tle at­ten­tion. Five years later, Al­bert Ein­stein (1905) 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. Nor­bert Wiener (1923) ul­ti­mately proved the ex­is­tence of Brown­ian mo­tion and de­vel­oped re­lated math­e­mat­i­cal the­o­ries, so Brown­ian mo­tion is often called a Wiener process.


Synonyms For Brownian Motion


Wiener process


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Definition Sources


Definitions for Brownian Motion are sourced/syndicated and enhanced from:

  • A Dictionary of Economics (Oxford Quick Reference)
  • Oxford Dictionary Of Accounting
  • Oxford Dictionary Of Business & Management

This glossary post was last updated: 3rd March, 2022 | 0 Views.