By far, the most interesting stochastic process used in financial economics is Brownian motion. The role of Brownian motion in stochastic processes is similar to that of Normal random variables in elementary statistics. The concept of a random walk, the discrete counterpart of the (continuous time) Brownian motion, is well known among students of economics, since most macroeconomic time series behave in a similar fashion (A random walk is a special case of what is known as unit root process or I(1) process). The plot given below shows trajectories (realizations) of a random walk process.

Common knowledge: The Brownian movement was discovered in 1827 by Robert Brown, a botanist. While he was studying microscopic live, he noticed little particles of plant pollens jiggling around in the liquid he was looking at in the microscope, and he was wise enough to realize that these were not living, but were just little pieces of dirt moving around in the water. In fact he helped to demonstrate that this had nothing to do with life by getting from the ground an old piece of quartz in which there was some water trapped. It must have been trapped for millions and millions of years, but inside he could see the same motion. What one sees is that very tiny particles are jiggling all the time. This was later proved to be one of the effects of molecular motion, and we can understand it qualitatively by thinking of a great push ball on a playing field, seen from a great distance, with a lot of people underneath, all pushing the ball in various directions. We cannot see the people because we imagine that we are too far away, but we can see the ball, and we notice that it moves around rather irregularly. [1] (Java Applet demonstrating 2-d Brownian motion).

In most financial calculus books Norbert Wiener and Albert Einstein are mentioned as the guys who provided the mathematical foundation for the theory of random motions. Of further historical interest is the fact that in his thesis, Théorie de la Spéculation, Bachelier (1900) proposed the Brownian motion as a model for stock prices*. In Bachelier's time, Paris was the centre of speculation in bonds and he reasoned that increments of stock prices should be idendependent and normally distributed. Since at that time any mathematical theory of the stock market was beneath the dignity of "real" mathematicians (his mentor was Henri Poincaré), Bachelier's work never received much attention.

WHAT YOU DID NOT KNOW: The earliest attempt to model Brownian motion mathematically can be traced to T.N. THIELE of Copenhagen, who effectively created a model of Brownian motion while studying time series in 1880 !!(Sur la compensation de quelques erreurs quasi-systématiques par la méthodes de moindre carrés, Reitzel, Copenhagen)[2]. Go to this site to download a copy of "Aspects of T.N. Thiele's contributions to statistics." (Bulletin of the International Statistical Institute).

*stock prices are often assumed to follow geometric Brownian motion, which is the same as saying that the logs of stock prices follow a random walk with drift.

[1] The Feynman Lectures on Physics, 1977, Volume 1
[2]"A Short history of Stochastic Integration and Mathematical Finance" by Robert Jarrow and Philip Protter.

R-code: randomwalk (txt, 0 KB)