Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 2

Brownian Motion

 

 

2.1. Definition and properties

Brownian motion plays a great role in mathematics, physics, biology, chemistry, and finance. First, it was used to describe the random chaotic movement of particles suspended in a liquid under bombardment by a tremendous number of atoms of the liquid.¹ The mass of the particle is much greater than that of a molecule, and therefore the influence of a single molecule blow is practically negligible. However, the number of blows is tremendous (about 10²¹ per second), and we can watch by microscope the uninterrupted chaotic movement of the particle. It is also important that all such blows are independent of each other. These facts lead to a mathematical model of Brownian motion. Mathematically, “chaotic” means that the trajectories of Brownian motion, though continuous, are nowhere differentiable. For this reason, a number of mathematical difficulties and “exotic” phenomena appear while considering questions related to Brownian motion.

Eventually it became clear that the Brownian motion is a much more universal random process. Many real life events are accompanied by a number of small independent factors (blows of “molecules”), each of which individually is probably completely irrelevant, but the overall sum of their effects can be quite perceptible. For example, consider stock fluctuations. We do not always see the reasons or shocks (“molecules”) that lead to chaotic changes of stock prices: political events and scandals, ...