The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Continuous-Time Stochastic Processes

  • Chi-Fu Huang
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_559

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 ts; (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.
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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Chi-Fu Huang
    • 1
  1. 1.