2218 CHAPTER 64. WIENER PROCESSES

Definition 64.6.7 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).

Proposition 64.6.8 The process defined in 64.6.37 is a Q Wiener process in H where Q =JJ∗.

Proof: First, why does the sum converge? Consider the sum for an increment in time.Let ti−1 = 0 to obtain the convergence of the sum for a given t. Consider the difference oftwo partial sums.

E

(n

∑k,l=m

(ψk (ti)−ψk (ti−1))Jgk,(ψ l (ti)−ψ l (ti−1))Jgk

)

= E

(n

∑k,l=m

(J∗Jgk,gl)(ψk (ti)−ψk (ti−1))(ψ l (ti)−ψ l (ti−1))

)

=n

∑k,l=m

(J∗Jgk,gl)E ((ψk (ti)−ψk (ti−1))(ψ l (ti)−ψ l (ti−1)))

=n

∑k=m

(J∗Jgk,gk)E(

ψk (ti)−ψk (ti−1)2)=

n

∑k=m

(J∗Jgk,gk)(ti− ti−1)

=n

∑k=m|Jgk|2H (ti− ti−1)

and this converges to 0 as m,n→ ∞ since J is Hilbert Schmidt. Thus the sum converges inL2 (Ω,H). Why are the disjoint increments independent?