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What is the “Brownian motion” and what is a Wiener process. History, importance, definition and applications (Bachelier, Wiener, Einstein, …):
Brownian Motion
Brownian motion refers to either the physical phenomenon that minute particles immersed in a fluid move around randomly or the mathematical models used to describe those random movements.[1]
Brownian motion was discovered by the biologist Robert Brown in 1827.
While Brown was studying pollen particles floating in water in the microscope, he observed minute particles in the pollen grains executing the jittery motion.
After repeating the experiment with particles of dust, he was able to conclude that the motion was due to pollen being “alive” but the origin of the motion remained unexplained.
The first one to give a theory of Brownian motion was Louis Bachelier in 1900 in his PhD thesis “The theory of speculation”. However, it was only in 1905 that Albert Einstein, using a probabilistic model, could sufficiently explain Brownian motion. He observed that if the kinetic energy of fluids was right, the molecules of water moved at random.
Thus, a small particle would receive a random number of impacts of random strength and from random directions in any short period of time. This random bombardment by the molecules of the fluid would cause a sufficiently small particle to move exactly just how Brown described it.
Wiener process
In mathematics, Brownian motion is described by the Wiener process, a continuous-time stochastic process named in honor of Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics.
The Wiener process Wt is characterized by four facts:
- W0 = 0
- Wt is almost surely continuous
- Wt has independent increments
- Wt-Ws ~ N(0,t-s)(for 0<= s<=t) Brownian motion is a well-suited model for a wide range of real random phenomena, from chaotic oscillations of microscopic objects, such as flower pollen in water, to stock market fluctuations. It is also a purely abstract mathematical tool which can be used to prove theorems in “deterministic” fields of mathematics.