Robert Brown Brownian Motion

Introduction to Brownian motion:

The movement of particle (random) suspended in a fluid is called Brownian motion. Brownian motion’s mathematical model has several applications in real world. One of which is the stock market fluctuations discussed below in this article. It is one of the simplest time-continuous probabilistic processes. This theory is closely related to that of normal distribution. In both of the theory, their motivation depends on mathematical convenience instead of the accuracy of models.

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History of Brownian motion


Thorvald N. Thiele first described Mathematics of Brownian motion in 1880 which was followed by Louis Bachelier’s thesis ‘the speculation theory’ in 1900s which contains stochastic analysis of stock and option markets.

A prediction made by Einstein that, at thermodynamic temperature T, Brownian motion of a particle in a fluid is characterized by a diffusion coefficient.

`D = (k_(B)T)/b`

kB is the Boltzmann's constant and b being the particle’s linear drag coefficient. After a time t, the root mean square displacement in any direction is

`sqrt(2) Dt`


Theory of Brownian motion


A one dimensional model which describes a particle that usually undergoes Brownian motion was published in 1906 by Smoluchowski. The collisions are assumed with M>>m (M is actually the mass of the test particle, whereas m is the mass of any of the singleton particles that composes the fluid). After collision the test particle’s velocity will increase by ?V˜ (m/M) v if V is test particles velocity and fluid particle velocity is v. If the number of collisions from the right is NR and the number of collisions from the left is NL, then the changed particle velocity after N collisions is given by ?V (2NR –N). The multiplicity is then given by:

`(N!) / (N_(R)!) (N - N_(R))!`

And total possible states are 2N.  Thus the probability that the particle will hit from the right NR times is

`P_(N)(N_(R)) = (N!) / (2^(N) N_(R)!) (N - N_(R))!`

Unfortunately, 1D model of Smoluchowski's can only explain Brownian motion more qualitatively Assumptions cannot be made for a real particle that undergoes the Brownian motion.

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Application of Brownian motion to market analysis


It used for the background information.

Brownian motion is used in stock market.

In Optical dynamic trading along with Leverage constraints.

In Price stabilization schemes, investments and uncertainty.

A Brownian motion model is used for decision making.