64.6. WIENER PROCESSES, ANOTHER APPROACH 2225

Therefore, ∫A

E(|W (t)|2− t tr(Q) |Fs

)−(|W (s)|2− s tr(Q)

)dP = 0

and since A ∈Fs is arbitrary, this shows{|W (t)|2− t tr(Q)

}is a martingale. Hence the

Doob Meyer decomposition for |W (t)|2 is

|W (t)|2 = Y (t)+ t tr(Q)

where Y (t) is a martingale.There is a generalization of Levy’s theorem to Hilbert space valued Wiener processes.

Theorem 64.6.10 Let {W (t)} ∈M 2T (H) ,E (W (t)) = 0, where H is a real separable Hil-

bert space. Then for Q a nonnegative symmetric trace class operator, {W (t)} is a Q Wiener

process if and only if both {W (t)} and{(W (t) ,h)2− t (Qh,h)

}are martingales for every

h ∈ H.

Proof: First suppose {W (t)} is a Q Wiener process. Then defining the filtration to be

Ft ≡ σ (W (s)−W (u) : u≤ s≤ t) ,

it follows from Lemma 64.4.2 that {W (t)} is a martingale. Consider{(W (t) ,h)2− t (Qh,h)

}.

Let A ∈Fs where s≤ t. Then using the fact {W (t)} is a martingale,∫A

E((W (t)−W (s) ,h)2 |Fs

)dP

=∫

AE((W (t) ,h)2 +(W (s) ,h)2−2(W (t) ,h)(W (s) ,h) |Fs

)dP

=∫

AE((W (t) ,h)2 |Fs

)+(W (s) ,h)2−E (2(W (t) ,h)(W (s) ,h) |Fs)dP

=∫

AE((W (t) ,h)2 |Fs

)dP+

∫A(W (s) ,h)2 dP

−∫

A(W (s) ,h)E (2(W (t) ,h) |Fs)dP

=∫

AE((W (t) ,h)2 |Fs

)dP−

∫A(W (s) ,h)2 dP.

Also since XA is independent of (W (t)−W (s) ,h)2 as in the proof of Lemma 64.4.2, and{W (t)} is a Q Wiener process, ∫

AE((W (t)−W (s) ,h)2 |Fs

)dP

=∫

A(W (t)−W (s) ,h)2 dP

= P(A)∫

Ω

(W (t)−W (s) ,h)2 dP

= P(A)(t− s)(Qh,h) .