ABSTRACT
Let us start by introducing a generalization of Brownian motion. Let F(x) be a continuous distribution function; and let w(t), 0 ≤ t ≤ 1, denote a standard Brownian motion. Then
W (x) = w◦F(x) = w(F(x)) is called a Brownian motion with respect to time F(x). In more detail, the standard Brownian motion on [0,1] is a zero-mean Gaussian process with independent increments. That is, for any k and any collection of k points 0 = t0 < t1 < · · ·< tk < tk+1 = 1 the increments
∆w(t j) = w(t j+1)−w(t j), j = 0, . . . ,k, are independent Gaussian random variables. The distribution of each ∆w(t j) has expected value 0 and variance ∆t j = t j+1− t j. Therefore, w(0) is identically zero, as it has expected value 0 and variance also 0. The distribution of w(t), which can also be regarded as the increment w(t) = w(t)−w(0), has expected value 0 and variance
Ew2(t) = t.
