8-Stochastic Integrals Page 230 Wednesday, February 4, 2004 12:50 PM
230 The Mathematics of Financial Modeling and Investment Management(i.e., the square of the spaced interval and the time interval are of the
same order) then the sequence of random walks approaches a Brownian
motion. Though this is intuitive as the binomial distributions approach
normal distributions, it should be clear that it is far from being mathe-
matically obvious.
Exhibit 8.1 illustrates 100 realizations of a Brownian motion
approximated as a random walk. The exhibit clearly illustrates that the
standard deviation grows with the square root of the time as the vari-
ance grows linearly with time. In fact, as illustrated, most paths remain
confined within a parabolic region.PROPERTIES OF BROWNIAN MOTION
The paths of a Brownian motion are rich structures with a number of
surprising properties. It can be demonstrated that the paths of a canoni-
cal Brownian motion, though continuous, are nowhere differentiable. It
can also be demonstrated that they are fractals of fractal dimension ³₂.EXHIBIT 8.1 Illustration of 100 Paths of a Brownian Motion Generated as an
Arithmetic Random Walk