Probability and Stochastic Processes

Chapter 15Brownian Motion

In this chapter we introduce—without a doubt—the most important stochastic process. This particular process is a Markov process, is a martingale, and, due to the martingale representation theorem, one of the most important in the theory of continuous-time–continuous-state space processes.

15.1 History

In 1827, the biologist, Robert Brown, looking through a microscope at pollen grains in water, noted that the grains moved through the water even though the stand was perfectly stationary. Repeating the experiment with inorganic matter, he disproved the hypothesis that the movement is caused by life matter; however, he was not able to determine the exact cause of the movement. This discovery led to speculation about atoms and molecules long before they were discovered. Of course, now we know that in fact the movement is caused by bombardment from the water molecules. Even though a pollen grain is 10,000 times larger than a water molecule, the cumulated effect of them is the driving force of the grain. The direction of the force of atomic bombardment is constantly changing, and at different times the pollen grain is hit more on one side than another, leading to the seemingly random nature of the motion. It is the fact that, even though the chances are that the pollen is hit equally from all sides, due to the random nature of life, this never happens in reality and this is the great discovery behind stochastic processes. This type of phenomenon is named ...