174 CHAPTER 5. BROWNIAN MOTION
5.4 Transformations of the Wiener Process
Rating
Mathematically Mature: may contain mathematics beyond calculus with
proofs.
Section Starter Question
Suppose you know the graphy =f(x) of the functionf(x). What is the
effect on the graph of the transformationf(x+h)−f(h)? What is the
effect on the graph of the transformationf(1/x)? Consider the function
f(x) = sin(x) as an example.
Key Concepts
- Three transformations of the Wiener process produce another Wiener
process. The transformations are scaling, inversion and translation.
These results prove especially helpful when studying the properties of
the sample paths of Brownian motion.
Vocabulary
- Scaling, also calledre-scaling, is the transformation off(t) tobf(t/a),
which expands or contracts the time axis (asa >1 ora <1) and ex-
pands or contracts the dependent variable scale (asb >1 orb <1). - Translation, also calledshiftingis the transformation off(t) tof(t+
h) or sometimesf(t) tof(t+h)−f(h). - Inversionis the transformation off(t) tof(1/t). It “flips” the inde-
pendent variable axis about 1, so that the interval (0,1) is “inverted”
to the interval (1,∞).