63.6.4 Levy’s Theorem In Hilbert Space
Recall the concept of quadratic variation. Let W
Wiener process. Does it
? The Wiener process is continuous. Furthermore,
for each t ∈
is a martingale, Theorem
can be applied to
Therefore, by the Doob Meyer decomposition, Theorem
, there exists an increasing natural process, A
and a martingale,
What is A
? Consider the process
From Theorem 63.5.4 this equals
is a one dimensional Wiener process and
By Lemma 63.4.2,
is a martingale. Therefore, for
s < t
and A ∈ℱs,
is independent of W
as in the proof of Lemma
and since A ∈ℱs is arbitrary, this shows
is a martingale. Hence the
Doob Meyer decomposition for
is a martingale.
There is a generalization of Levy’s theorem to Hilbert space valued Wiener
Theorem 63.6.10 Let
, where H is a
real separable Hilbert space. Then for Q a nonnegative symmetric trace class
is a Q Wiener process if and only if both
are martingales for every h ∈ H.
Proof: First suppose
Wiener process. Then defining the filtration to
it follows from Lemma 63.4.2 that
is a martingale. Consider
Let A ∈ℱs where s ≤ t. Then using the fact
is a martingale,
Also since XA is independent of
as in the proof of Lemma 63.4.2
Thus, this has shown that for all A ∈ℱs,
and since A ∈ℱs
is arbitrary, this proves
This proves one half of the theorem.
Next suppose both
are martingales for any
h ∈ H.
It follows that both
also. Therefore, by Levy’s theorem, Theorem
is a Wiener
process with the property that its variance at
instead of t.
Thus the time increments are normal and independent. I need to verify that
Wiener process. One of the things which needs to be shown is
I have just shown
which follows from Levy’s theorem which concludes
is a Wiener process.
Now using 63.6.42
, it follows from this that
which shows 63.6.41. This completes the proof.