Mathematical Modeling in Finance with Stochastic Processes

176 CHAPTER 5. BROWNIAN MOTION

(c)

Wshift(0) =W(0 +h)−W(h) = 0.

(d) As the composition and difference of continuous functions,Wshift

is continuous.

2.

Wscale(t) =cW(t/c^2 )

(a) The increment

Wscale(t)−Wscale(s) =cW((t)/c^2 )−cW(s/c^2 ) =c(W(t/c^2 )−W(s/c^2 ))

is normally distributed because it is a multiple of a normally dis-

tributed random variable. Since the incrementW(t/c^2 )−W(s/c^2 )

has mean zero, then

Wscale(t)−Wscale(s) =c(W(t/c^2 )−W(s/c^2 ))

must have mean zero. The variance is

E

[

(Wscale(t)−W(s))^2

]

=E

[

(cW((t)/c^2 )−cW(s/c^2 ))^2

]

=c^2 E

[

(W(t/c^2 )−W(s/c^2 ))^2

]

=c^2 (t/c^2 −s/c^2 ) =t−s.

(b) Note that ift 1 < t 2 ≤ t 3 < t 4 , thent 1 /c^2 < t 2 /c^2 ≤ t 3 /c^2 <

t 4 /c^2 , and the corresponding incrementsW(t 4 /c^2 )−W(t 3 /c^2 ) and

W(t 2 /c^2 )−W(t 1 /c^2 ) are independent. Then the multiples of each

bycare independent and soWscale(t 4 )−Wscale(t 3 ) andWscale(t 2 )−

Wscale(t 1 ) are independent.

(c) Wscale(0) =cW(0/c^2 ) =cW(0) = 0.

(d) As the composition of continuous functions,Wscaleis continuous.

Theorem 16.SupposeW(t)is a Standard Wiener Process. Then the trans-
formed processesWinv(t) =tW(1/t)for t > 0 ,Winv(t) = 0for t= 0is a
version of the Standard Wiener Process.