64.5. HILBERT SPACE VALUED WIENER PROCESSES 2201

= P(A)∫

Ω

||W (t)−W (s)||2− (t− s) traceQdP

+∫

A

(2(W (t) ,W (s))−||W (s)||2− trace(Q)s|Fs

)dP

=∫

A2(W (s) ,E (W (t) |Fs))dP−

∫A||W (s)||2 dP−

∫A

s traceQdP

=∫

A

(||W (s)||2− s traceQ

)dP

and this shows that the quadratic variation [W ] (t) = t trace(Q) by uniqueness of the quad-ratic variation.

Now suppose you start with a nonnegative trace class operator Q. Then in this case alsoone can define a Q Wiener process. It is possible to get this theorem from Theorem 64.5.2but this will not be done here.

Theorem 64.5.3 Let U be a real separable Hilbert space and let Q be a nonnegative traceclass operator defined on U. Then there exists a Q Wiener process as defined in Definition64.5.1. Furthermore, the distribution of W (t)−W (s) is the same as the distribution ofW (t− s) and W is Holder continuous with exponent γ for any γ < 1/2.

Proof: One can obtain this theorem as a corollary of Theorem 64.5.2 but this will notbe done here.

Let

Q =∞

∑i=1

λ iei⊗ ei

where {ei} is a complete orthonormal set and λ i ≥ 0 and ∑λ i < ∞. Now the definition ofthe Q Wiener process is

W (t)≡∞

∑k=1

√λ kekψk (t) (64.5.21)

where {ψk (t)} are the real Wiener processes defined in Lemma 64.3.1.Now consider 64.5.21. From this formula, if s < t

W (t)−W (s) =∞

∑k=1

√λ kek (ψk (t)−ψk (s)) (64.5.22)

First it is necessary to show this sum converges. Since ψ j (t) is a Wiener process,

∫Ω

∣∣∣∣∣ n

∑j=m

√λ j

(ψ j (t)−ψ j (s)

)e j

∣∣∣∣∣2

U

dP

=∫

Ω

n

∑j=m

λ j

(ψ j (t)−ψ j (s)

)2dP

= (t− s)n

∑j=m

λ j