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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.
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.

Brownian Motion as a Limiting Case of a Random Walk
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. If you have not already done so, see the notation conventions documentation. 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 a realization, the result still looks like a realization of a Brownian motion. This is illustrated in Exhibit 2.

Realizations of a Brownian motion are fractals
Exhibit 2

Realizations of Brownian motions have the same jagged appearance no matter how much you magnify them.
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.
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—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.

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 made significant contributions to related mathematical theories, so Brownian motion is often called a Wiener process.

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