ABSTRACT
We continue to consider different types of random processes keeping the notation from Chapter 4.
In this section, we revisit Brownian motion, or the Wiener process, wt defined in Section 4.1.2.2.
1.1.1 Non-dierentiability of trajectories
The definition of wt in Section 4.1.2.2 requires the trajectories of the process to be continuous. Let us turn to differentiability. As before, let wD stand for the increment of wt and F(x) denote the standard normal d.f. To determine whether wt has a derivative at a point t, consider a time interval D= (t; t+d]
and explore the behavior of the r.v. hD = wD=d as d! 0. By definition, the r.v. wD is normal with zero mean and a standard deviation of
p d. Then
for x 0, we have P(jhDj > x) = P(jwDj > xd) = 2(1F(xd= p d)) = 2(1F(x
p d))!
2(1F(0)) = 1 as d ! 0. Since the last relation is true for an arbitrary large x, this means that when d is approaching zero, the r.v. jhDj takes on arbitrary large values with a probability close to one. Rigorously, jhDj ! ¥ as d! 0 in probability (for a definition of this type of convergence see Section 0.5). As a matter of fact, an even stronger property is true. Namely, with probability one,
trajectories of Brownian motion (that is, wt as a function of t) are nowhere differentiable; i.e., the derivative does not exist at any point. (See, e.g., [112, p.32] or the outline of a proof in [70, p.268].) This is an amazing property. Trajectories are continuous but not smooth, and the process
fluctuates infinitely frequently in any arbitrary small time interval. However, this is not an obstacle for applications. If we are interested in the increments of the process over intervals that perhaps are small but not infinitesimally small, then we are dealing with r.v.’s wD which are normal in the mathematical and usual sense as well, and hence are tractable.
