64.6. WIENER PROCESSES, ANOTHER APPROACH 2219

Let λ k ∈ H. Consider t0 < t1 < · · ·< tn.

E

(exp i

n

∑k=1

(λ k,W (tk)−W (tk−1))

)=

n

∏k=1

E (exp(i(λ k,W (tk)−W (tk−1))))? (64.6.38)

Start with the left. There are finitely many increments concerned and so it can be assumedthat for each k one can have m→ ∞ such that the partial sums up to m in the definition ofW (tk)−W (tk−1) converge pointwise a.e. Thus

E

(exp i

n

∑k=1

(λ k,W (tk)−W (tk−1))

)

= limm→∞

E

(exp i

n

∑k=1

(λ k,

m

∑j=1

(ψ j (tk)−ψ j (tk−1)

)Jg j

))

= limm→∞

E

(exp

n

∑k=1

m

∑j=1

i(

λ k,(

ψ j (tk)−ψ j (tk−1))

Jg j

))

= limm→∞

E

(m

∏j=1

exp

(n

∑k=1

i(

λ k,(

ψ j (tk)−ψ j (tk−1))

Jg j

)))

Now from 64.6.36,{

∑nk=1 i

(λ k,(

ψ j (tk)−ψ j (tk−1))

Jg j

)}m

j=1are independent. Hence

the above equals

= limm→∞

m

∏j=1

E

(exp

(n

∑k=1

i(

λ k,(

ψ j (tk)−ψ j (tk−1))

Jg j

)))

= limm→∞

m

∏j=1

E

(n

∏k=1

exp(

i(

λ k,(

ψ j (tk)−ψ j (tk−1))

Jg j

)))Now from independence of the increments for the ψ j, this equals

= limm→∞

m

∏j=1

n

∏k=1

E(

exp(

i(

λ k,(

ψ j (tk)−ψ j (tk−1))

Jg j

)))

= limm→∞

m

∏j=1

n

∏k=1

E(

exp(

i(λ k,Jg j)(

ψ j (tk)−ψ j (tk−1))))

= limm→∞

m

∏j=1

n

∏k=1

e−12 (λ k,Jg j)

2(tk−tk−1) = lim

m→∞

m

∏j=1

e−12 ∑

nk=1(λ k,Jg j)

2(tk−tk−1)

= limm→∞

exp

(−1

2

m

∑j=1

n

∑k=1

(λ k,Jg j)2 (tk− tk−1)

)

= exp

(−1

2

n

∑k=1

∞

∑j=1

(J∗λ k,g j)2 (tk− tk−1)

)(64.6.39)