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.
Let’s formalize this. A univariate Brownian motion is defined as a stochastic process B satisfying
- The process is defined for times t ≥ 0, with 0B = 0.
- Realizations are continuous functions of time t.
- Random variables tB – sB are normally distributed with mean 0 and variance t – s, for t > s.
- 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.