ABSTRACT
Brownian motion is a keystone in the foundation of mathematical finance. Brownian motion is referred as a continuous-time stochastic process but it can be constructed as a limiting case of symmetric random walks. Since the probability distribution of Brownian motion is normal, this chapter reviews some of the important basic facts about the multivariate normal distribution. It derives the transition probability distribution of Brownian motion. Geometric Brownian motion is the most well-known stochastic model for modelling positive asset price processes and pricing contingent claims. A bridge process is obtained by conditioning on the value of the original process at some future time. The chapter shows that realizations of the Brownian bridge have the same probability distribution as those of the Brownian motion. Most of the continuous-time processes belong to the class of Gaussian processes. The most well-known representative of such a class is Brownian motion.
