2222 CHAPTER 64. WIENER PROCESSES

Differentiate with respect to α

E (i( f ,W (t)−W (s))exp i(α f +βg,W (t)−W (s)))

= − [α (Q f , f )+(Q f ,βg)] (t− s)exp(−1

2(Q(α f +βg) ,α f +βg)(t− s)

)Let α = 0.

E (i( f ,W (t)−W (s))exp i(βg,W (t)−W (s)))

= − [(Q f ,βg)] (t− s)exp(−1

2(Q(βg) ,βg)(t− s)

)Now differentiate with respect to β

E (−( f ,W (t)−W (s))(g,W (t)−W (s))exp i(βg,W (t)−W (s)))

=− [(Q f ,g)] (t− s)exp(−1

2(Q(βg) ,βg)(t− s)

)+− [(Q f ,βg)] (t− s)(something)

Now let β = 0.

E (( f ,W (t)−W (s))(g,W (t)−W (s))) = (Q f ,g)(t− s)

Finally, Q = JJ∗. It is self adjoint and nonnegative and so there is a complete orthonor-mal basis {ei} such that Qei = λ iei. Then λ i = (Qei,ei)H and so

∑i

λ i = ∑i(Qei,ei) = ∑

i|J∗ei|2U < ∞

because J and hence J∗ are both Hilbert Schmidt operators.Recall the notion of the Hilbert space LU in Definition 19.2.1.What if you have a given Q ∈L (H,H) which is trace class,Q = Q∗, and nonnegative.

Does there exist a Q Wiener process of the sort just described? It appears this amounts toobtaining a Hilbert Schmidt map J from some Hilbert space U to H such that Q = JJ∗.

Since Q is trace class and is self adjoint, it follows that there is an orthonormal basis{ei} ,Qei = λ iei, where λ i is positive for i≤ L or positive for all i. Then

Q1/2 =L

∑i=1

√λ iei⊗ ei

andQ1/2ei =

√λ iei.

Then also on Q1/2H, (Q1/2ei,Q1/2e j

)Q1/2H

≡ (ei,e j)H

and so an orthonormal basis in Q1/2H is{√

λ iei}L

i=1. Then define J : Q1/2H→ H

Jx≡L

∑k=1

(x,√

λ kek

)Q1/2H

√λ kek