2224 CHAPTER 64. WIENER PROCESSES

64.6.4 Levy’s Theorem In Hilbert SpaceRecall the concept of quadratic variation. Let W (t) be a Q Wiener process. Does it follow{W (t)} ∈M 2

T (H)? The Wiener process is continuous. Furthermore,

E(|W (t)|2H

)< ∞

for each t ∈ [0,T ] . Since {W (t)} is a martingale, Theorem 62.5.3 can be applied to con-clude

E(|W (t)|2H

)1/2≤ E

( supt∈[0,T ]

|W (t)|)21/2

≤ 2E(|W (T )|2H

)1/2

and so {W (t)} ∈M 2T (H) . Therefore, by the Doob Meyer decomposition, which is The-

orem 63.7.15, there exists an increasing natural process, A(t) and a martingale, Y (t) suchthat

|W (t)|2H = Y (t)+A(t) .

What is A(t)? Consider the process|W (t)|2

From Theorem 64.5.4 this equals∞

∑k=1

λ kψk (t)2

where ψk (t) is a one dimensional Wiener process and

Q =∞

∑k=1

λ kek⊗ ek,∞

∑k=1

λ k < ∞.

By Lemma 64.4.2, {W (t)} is a martingale. Therefore, for s< t and A∈Fs, it follows sinceXA is independent of W (t)−W (s) as in the proof of Lemma 64.4.2 that the followingholds. ∫

AE(|W (t)|2 |Fs

)−|W (s)|2 dP

=∫

AE(|W (t)|2 + |W (s)|2−2W (t) ·W (s) |Fs

)dP

=∫

AE(|W (t)−W (s)|2 |Fs

)dP =

∫A|W (t)−W (s)|2 dP

= P(A)∫

Ω

|W (t)−W (s)|2 dP

= P(A)∞

∑k=1

λ kE((ψk (t)−ψk (s))

2)

= P(A)(t− s)∞

∑k=1

λ k = P(A)(t− s) tr(Q) .