Continuous-Time Stochastic Processes

Abstract

Applications of continuous-time stochastic processes to economic modelling are largely focused on the areas of capital theory and financial markets. In these applications as in mathematics generally, the most widely studied continuous time process is a Brownian motion – so named for its early application as a model of the seemingly random movements of particles which were first observed by the English botanist Robert Brown in the 19th century. Einstein (1905), in the context of statistical mechanics, is generally given credit for the first mathematical formulation of a Brownian motion process. However, an earlier development of an equivalent continuous-time process is provided by Louis Bachelier (1900) in his theory of stock option pricing. Framed as an abstract mathematical process, a Brownian motion $0\le s<t<\infty, B(t)-B(s)$ is a normally distributed random variable with mean zero and variance t – s; (2) for $0\le {t}_0<{t}_1<\cdots <{t}_1<\infty$,

$$ \left\{B\left({t}_0\right);B\left({t}_k\right)-B\left({t}_{k-1}\right),k=1,\dots, l\right\} $$

is a set of independent random variables.

This chapter was originally published in The New Palgrave: A Dictionary of Economics, 1st edition, 1987. Edited by John Eatwell, Murray Milgate and Peter Newman