2194 CHAPTER 64. WIENER PROCESSES

Why is C ([0,T ] ;E) separable? You can assume without loss of generality that theinterval is [0,1] and consider the Bernstein polynomials

pn (t)≡n

∑k=0

(nk

)f(

kn

)tk (1− t)n−k

These converge uniformly to f Now look at all polynomials of the form

n

∑k=0

aktk(

1− tk)

where the ak is one of the countable dense set and n ∈ N. Each Bernstein polynomialuniformly close to one of these and also uniformly close to f . Hence polynomials of thissort are countable and dense in C ([0,T ] ;E).

64.5 Hilbert Space Valued Wiener ProcessesNext I will consider the case of Hilbert space valued Wiener processes. This will includethe case of Rn valued Wiener processes. I will present this material independent of themore general case of E valued Wiener processes.

Definition 64.5.1 Let 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, whenever t1 <t2 < · · ·< tm, the increments {W (ti)−W (ti−1)} are independent, W (0) = 0, and whenevers < 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))

AlsoE ((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 =∞

∑i=1

λ iei⊗ ei

where {ei} is a complete orthonormal basis, λ i ≥ 0, and

∞

∑i=1

λ i < ∞

Such a stochastic process is called a Q Wiener process. In the case where these have valuesin Rn tQ ends up being the covariance matrix of W (t).