ABSTRACT
This chapter deals with the one-dimensional Brownian motion. The first approaches to mathematically modeling the Brownian motion were made by L. Bachelier and A. Einstein. Brownian motion has fruitful applications in key disciplines as time series analysis, operations research, communication theory, and reliability theory. The Brownian motion, as any Gaussian process, is completely determined by its trend and covariance function. Actually, since the trend function of a Brownian motion is identically zero, the Brownian motion is completely characterized by its covariance function. Both the Brownian bridge and the geometric Brownian bridge have some significance in modelling stochastically fluctuating parameters in mathematics of finance. The Ornstein-Uhlenbeck process, as the Brownian motion, is a Markov process. Brownian motion processes with drift are, amongst other applications, used for modeling financial parameters, productivity criteria, cumulative maintenance costs, wear modeling as well as for modeling physical noise.
