Probability Theory pp 447-494 | Cite as

# Brownian Motion

In Example 14.45, we constructed a (canonical) process (*X*_{ t })_{t∈[0,∞)} with independent stationary normally distributed increments. For example, such a process can be used to describe the motion of a particle immersed in water or the change of prices in the stock market. We are now interested in properties of this process *X* that cannot be described in terms of finite-dimensional distributions but reflect the whole *path t* → *X*_{ t }. For example, we want to compute the distribution of the functional *F*(*X*) := *sup*_{t∈[0,1]} *X*_{ t }. The first problem that has to be resolved is to show that *F* is a random variable.

In this chapter, we investigate continuity properties of paths of stochastic processes and show how they ensure measurability of some path functionals. Then we construct a version of *X* that has continuous paths, the so-called *Wiener process* or *Brownian motion*. Without exaggeration, it can be stated that Brownian motion is *the* central object of probability theory.

## Keywords

Brownian Motion Continuous Path Local Martingale Brownian Bridge Path Property## Preview

Unable to display preview. Download preview PDF.