Brownian Motion

Brownian motion is both the irregular physical motion of microscopic particles suspended in a fluid and a mathematical stochastic process used to model such motion. The name honors the botanist Robert Brown, who observed the phenomenon in 1827 and reported it the following year (Brown 1828). The mathematical term refers to an idealized stochastic process rather than the motion of any particular particle.

A Brownian motion with initial value is a real-valued stochastic process that satisfies the following properties:

1. .

2. For all times , the increments , , ..., , are independent random variables.

3. For all , , the increments are normally distributed with expectation value zero and variance .

4. The function is continuous.

A Brownian motion is said to be standard if .

A geometric Brownian motion is obtained by exponentiating a standard Brownian motion together with a linear drift. In particular, has multiplicative random changes while is a scaled Brownian motion with drift.

It is easily shown from the above criteria that a Brownian motion has a number of unique natural invariance properties including scaling invariance and invariance under time inversion. Moreover, any Brownian motion satisfies a law of large numbers so that

(1)

almost surely. Moreover, for every , a Brownian path is locally Hölder continuous almost surely. This means that, on each finite time interval, there is a random constant such that

(2)

for all and in the interval. On the other hand, a Brownian path is nowhere differentiable almost surely.

The above definition is extended naturally to get higher-dimensional Brownian motions. More precisely, given independent Brownian motions , ..., which start at , ..., , one can define a stochastic process by

(3)

Such a is called a -dimensional Brownian motion which starts at .