Next I will consider the case of Hilbert space valued Wiener processes. This will include
the case of ℝ^{n} valued Wiener processes. I will present this material independent of the
more general case of E valued Wiener processes.
Definition 63.5.1Let W
(t)
be a stochastic process with values in H, a real separableHilbert space which has the properties that t → W
(t,ω)
is continuous, whenevert_{1}< t_{2}<
⋅⋅⋅
< t_{m}, the increments
{W (ti) − W (ti−1)}
are independent, W
(0)
= 0, andwhenever s < t,
ℒ (W (t) − W (s)) = N (0,(t− s)Q)
which means that whenever h ∈ H,
ℒ ((h,W (t)− W (s))) = N (0,(t− s)(Qh,h ))
Also
E ((h1,W (t) − W (s))(h2,W (t)− W (s))) = (Qh1,h2) (t− s).
Here Q is a nonnegative trace class operator. Recall this means
∑∞
Q = λiei ⊗ ei
i=1
where
{ei}
is a complete orthonormal basis, λ_{i}≥ 0, and
∞∑
λi < ∞
i=1
Such a stochastic process is called a Q Wiener process.In the case where these havevalues in ℝ^{n}tQ ends up being the covariance matrix of W
(t)
.
Note the characteristic function of a Q Wiener process is
( i(h,W (t))) − 12t2(Qh,h)
E e = e (63.5.18)
(63.5.18)
Note that by Theorem 60.8.5 if you simply say that the distribution measure of W
(t)
is Gaussian, then it follows there exists a trace class operator Q_{t} and m_{t}∈ H such that
this measure is N
(mt,Qt)
. Thus for W
(t)
a Wiener process, Q_{t} = tQ and m_{t} = 0. In
addition, the increments are independent so this is much more specific than the earlier
definition of a Gaussian measure.
What is a Q Wiener process if the Hilbert space is ℝ^{n}? In particular, what is Q? It is
given that
ℒ ((h,W (t)− W (s))) = N (0,(t− s)(Qh,h))
In this case everything is a vector in ℝ^{n} and so for h ∈ ℝ^{n},
( ) 2
E eiλ(h,W (t)− W(s)) = e− 12λ(t− s)(Qh,h)
In particular, letting λ = 1 this shows W
(t)
− W
(s)
is normally distributed with
covariance
(t− s)
Q because its characteristic function is e^{−1
2
h∗(t− s)
Qh
}.
With this and definition, one can describe Hilbert space valued Wiener processes in a
fairly general setting.
Theorem 63.5.2Let U be a real separable Hilbert space and let J : U_{0}→ U be a HilbertSchmidt operator where U_{0}is a real separable Hilbert space. Then let
{gk}
be a completeorthonormal basis for U_{0}and define for t ∈
[0,T]
∑∞
W (t) ≡ ψk (t)Jgk
k=1
Then W
(t)
is a Q Wiener process for Q = JJ^{∗}as in Definition 63.5.1. Furthermore, thedistribution of W
(t)
− W
(s)
is the same as the distribution of W
(t− s)
, and W isHolder continuous with exponent γ for any γ < 1∕2. There also is a subsequence denotedby N such that the convergence of the series
N∑
ψk(t)Jgk
k=1
is uniform for all ω not in some set of measure zero.
Proof: First it is necessary to show the series converges in L^{2}
(Ω;U )
for each t. For
convenience I will consider the series for W