Brownian Motion

This chapter discusses the basics of the Brownian motion by introducing the essential properties of the process. Brownian motion is an important building block for modeling continuous-time stochastic processes. In particular, it has become an important framework for modeling financial markets. There are many reasons for studying the Brownian motion. It is an important building block for modeling continuous-time stochastic processes because many classes of stochastic processes contain Brownian motion. It is a Markov process, a Gaussian process, a martingale, a diffusion process, as well as a Levy process. Over the years it has become a rich mathematical object. Langevin equation, which is the most widely known mathematical model of Brownian motion, is used to obtain the mean-square displacement (MSD) of the process.