Chapter 1
One-Dimensional
Brownian Motion
1.1. Some motivation
The biologist Robert Brown noticed almost two hundred years ago that bits
of pollen suspended in water undergo chaotic behavior. The bits of pollen
are much more massive than the molecules of water, but of course there are
many more of these molecules than there are bits of pollen. The chaotic
motion of the pollen is the result of many infinitesimal jolts by the water
molecules. By the central limit theorem (CLT), the law of the motion of
the pollen should be closely related to the normal distribution. We now call
this law Brownian motion.
During the past half century or so, Brownian motion has turned out
to be a very versatile tool for both theory and applications. As we will
see in Chapter 6, it provides a very elegant and general treatment of the
Dirichlet problem, which asks for harmonic functions on a domain with
prescribed boundary values. It is also the main building block for the theory
of stochastic calculus, which is the subject of Chapter 5. Via stochastic
calculus, it has played an important role in the development of financial
mathematics.
As we will see later in this chapter, Brownian paths are quite rough
they are of unbounded variation in every time interval. Therefore, integrals
with respect to them cannot be defined in the Stieltjes sense. A new type
of integral must be defined, which carries the name of K. Itˆ o, and more
recently, of W. Doeblin. This new integral has some unexpected properties.
Here is an example: If B(t) is standard Brownian motion at time t with
1
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