2198 CHAPTER 64. WIENER PROCESSES

Next consider the claim that the increments are independent. Let W N (t) be given bythe appropriate partial sum and let

{h j}m

j=1 be a finite list of vectors of U . Then from theindependence properties of ψ j explained above,

E

(exp

m

∑j=1

i(h j,W N (t j)−W N (t j−1

))U

)

E

(exp

m

∑j=1

i

(h j,

N

∑k=1

Jgk(ψk (t j)−ψk

(t j−1

)))U

)

= E

(exp

m

∑j=1

N

∑k=1

i(h j,Jgk)U(ψk (t j)−ψk

(t j−1

)))

= E

(∏j,k

exp(i(h j,Jgk)U

(ψk (t j)−ψk

(t j−1

))))

= ∏j,k

E(exp(i(h j,Jgk)U

(ψk (t j)−ψk

(t j−1

))))

This can be done because of the independence of the random variables

{ψk (t j)−ψk

(t j−1

)}j,k .

Thus the above equals

∏j,k

exp(−1

2(h j,Jgk)

2U

(t j− t j−1

))

=m

∏j=1

exp

(−1

2

N

∑k=1

(h j,Jgk)2U

(t j− t j−1

))

because ψk (t j)−ψk(t j−1

)is normally distributed having variance t j− t j−1. Now letting