Mathematical Modeling in Finance with Stochastic Processes

5.7 Quadratic Variation of the Wiener Process

Mathematical Ideas

Transformations of the Wiener Process

A set of transformations of the Wiener process produce the Wiener process
again. Since these transformations result in the Wiener process, each tells
us something about the “shape” and “characteristics” of the Wiener process.
These results prove especially helpful when studying the properties of the
Wiener process sample paths. The first of these transformations is a time
homogeneity which says that the Wiener process can be re-started anywhere.
The second says that the Wiener process can be rescaled in time and space.
The third is an inversion. Roughly, each of these says the Wiener process
is self-similar in various ways. See the comments after the proof for more
detail.

Theorem 2. 1. Wshift(t) =W(t+h)−W(h), for fixedh >0.

2. Wscale(t) =cW(t/c^2 ), for fixedc > 0
   are each a version of the Standard Wiener Process.

Proof. We have to systematically check each of the defining properties of the
Wiener process in turn for each of the transformed processes.

1.

Wshift(t) =W(t+h)−W(h)

(a) The increment

Wshift(t+s)−Wshift(s) = [W(t+s+h)−W(h)]−[W(s+h)−W(h)] =W(t+s+h)−W(s+h)

which is by definition normally distributed with mean 0 and vari-

ancet.

(b) The increment

Wshift(t 4 )−Wshift(t 3 ) =W(t 4 +h)−W(t 3 +h)

is independent from

W(t 2 +h)−W(t 1 +h) =Wshift(t 2 )−Wshift(t 1 ),

by the property of independence of disjoint increments ofW(t).