Abstract
Brownian motion is a continuous-time process that is used to model the uncertainty in the motion of “particles” in different media. The process is named after the Scottish botanist Robert Brown who in 1828 observed that a pollen particle suspended in a fluid continually moved randomly on the surface. Einstein (1905) suggested that the movements were due to collisions between the liquid’s molecules and the relatively light pollen. Brownian motion can also be thought of as an “infinitesimal random walk” in which smaller and smaller steps over extremely small time intervals, which is how it often arises in many applications. Thus, it can be defined as the limit of a random walk when the time steps and the jump sizes go to zero. Although Brownian motion is everywhere continuous, it is nowhere differentiable because Brownian paths are quite rough. In fact, no matter how small a segment of the path is chosen, you will never find a segment that is a straight line. Thus, a new type of integral called the Ito integral was defined to handle the stochastic differential equation (SDE) for the Brownian motion. The process is a fractal because you can zoom in on its sample path under magnification and the jaggedness never seems to smooth out. Brownian motion is used as a building block for models in several disciplines. It has been successfully used to describe thermal noise in electrical circuits, limiting behavior of queueing networks under heavy traffic, population dynamics in biological systems, and economic processes such as stock prices. In this chapter, we discuss different aspects of Brownian motion, including the geometric Brownian motion (GBM), Brownian bridge, fractional Brownian motion (FBM), Ornstein–Uhlenbeck (OU) process, and SDE.
Published Version
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