64.6. WIENER PROCESSES, ANOTHER APPROACH 2217

You could also take an arbitrary f ∈ L2 (0,∞) and consider W (t) ≡W(X(0,t) f

). You

could consider this as an integral and write it in the notation

W (t)≡∫ t

0f dW ≡W

(f X(0,t)

)Then from the construction,

E

((∫ t

0f dW

)2)

= E(

W(

f X(0,t))2)=∫ T

0f 2X(0,t)ds =

∫ t

0| f |2 ds = E

(∫ t

0| f |2 ds

)because f does not depend on ω . This of course is formally the Ito isometry.

64.6.3 Q Wiener Processes In Hilbert SpaceNow let U be a real separable Hilbert space. Let an orthonormal basis for U be {gi}. Nowlet L2 (0,∞,U) be H in the above construction. For h,g ∈ L2 (0,∞,U) .

E (W (h)W (g)) = (h,g)L2(0,∞,U) ≡ (h,g)H

Here each W (g) will be a real valued normal random variable, the variance of W (g) is|g|2L2(0,∞,U) and its mean is 0, every vector (W (h1) , · · · ,W (hn)) being generalized multi-variate normal. Let

ψk (t) =W(X(0,t)gk

).

Then this is a real valued random variable. Disjoint increments are obviously independentin the same way as before. Also

E(

ψk (t)ψ j (s))= E

(W(X(0,t)gk

)W(X(0,s)g j

))≡∫

∞

0X(0,t∧s) (gk,g j)U dt = 0

(64.6.36)if j ̸= k. Thus the random variables ψk (t) and ψ j (s) are independent. This is because, from

the construction,(

ψk (t) ,ψ j (s))

is normally distributed and the covariance is a diagonalmatrix. Also

ψk (t)−ψk (s) =W(X(0,t)Jgk

)−W

(X(0,s)Jgk

)=W

(X(s,t)Jgk

)ψk (t− s)≡W

(X(0,t−s)Jgk

)so ψk (t− s) has the same mean, 0 and variance, |t− s| , as ψk (t)−ψs (s). Thus these havethe same distribution because both are normally distributed.

Now let J be a Hilbert Schmidt map from U to H. Then consider

W (t) = ∑k

ψk (t)Jgk. (64.6.37)

This has values in H. It is shown below that the series converges in L2 (Ω;H). Recall thedefinition of a Q Wiener process.