Chapter 68

A Different Kind Of Stochastic Integra-tion

For more on this material, see [102] which is what this is based on. Recall the followingcorollary. It is Corollary 59.20.3 on Page 1936.

Corollary 68.0.1 Let H be a real Hilbert space. Then there exist random variables W (h)for h ∈ H such that each is normally distributed with mean 0 and for every h,g, it followsthat (W (h) ,W (g)) is normally distributed and

E (W (h)W (g)) = (h,g)H

Furthermore, if {ei} is an orthogonal set of vectors of H, then {W (ei)} are independentrandom variables. Also for any finite set { f1, f2, · · · , fn},

(W ( f1) ,W ( f2) , · · · ,W ( fn))

is normally distributed.

Here are some simple examples.

Example 68.0.2 Let H = L2 ([0,T ]) . For f ∈ H, let

W ( f )≡∫ T

0f (u)dW

where W (t) is the one dimensional Wiener process.

First of all, note that the integrand is adapted to the usual filtration determined by theWiener process. This is because f does not depend on ω. That W ( f ) is normally distributedcan be seen from the approximation of the Ito integral with the integral of elementaryfunctions. These are clearly normally distributed because they are just linear combinationsof increments of the Wiener process. Recall these increments were independent. Thus theintegrals of these elementary functions are all normally distributed. If In (ω) is one of these,then In→ I in L2 (Ω) where I is the above Ito integral. It follows that

E(eiIt)= lim

n→∞E(eiInt)= lim

n→∞e−(1/2)∥ fn∥2L2 t = e−(1/2)∥ f∥2

L2 t

so in fact, W ( f ) is normally distributed with mean 0 and variance ∥ f∥2L2 . As to the other

condition, the Ito isometry implies that

E (W ( f )W (g)) =∫ T

0f (u)g(u)du = ( f ,g)H

One can verify this by considering E(

W ( f +g)2),E(

W ( f −g)2)

.This example is called the isonormal Gaussian process. There is a measure space

(Ω,F ,P) where σ (W (s) : s≤ T )≡F . There must be an underlying measure space whichcomes from having to define the Wiener process.

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