2300 CHAPTER 67. THE EASY ITO FORMULA

67.9 Some Representation TheoremsIn this section is a very interesting representation theorem which comes from the Ito for-mula. In all of this, W will be a Q Wiener process having values in Rn for which Q = I.Recall that, letting

Gt ≡ σ (W(s) : s≤ t)

the normal filtration determined by the Wiener process is given by

Ft ≡ ∩s>tGs

where Gs is the completion of Gs. In this section, the theorems will all feature the smallerfiltration Gt , not the filtration Ft . First here are some simple observations which tie thisspecialized material to what was presented earlier.

When you have f an Gt adapted function in L2 (Ω,Rn) , you can consider

fT ∈ L2([0,T ]×Ω;L2

(Q1/2Rn,R

))as follows. Letting {gi} be an orthonormal basis for the subspace Q1/2Rn in the norm ofQ1/2Rn,

∥f∥2L2(Q1/2Rn,R) ≡∑

i

(fT gi

)2< ∞

For simplicity, let Q = I. Then you have the simple situation that∥∥fT∥∥L2(Q1/2Rn,R) = ∥f∥

2Rn

In what follows Wt will be the Q Wiener process onRn where Q = I. Then the Ito isometryis nothing more than the following lemma.

Lemma 67.9.1 Let f be Ft adapted in the sense that every component is Ft adapted andf ∈ L2 (Ω;Rn). Here Ft is the normal filtration coming from the Wiener process. Then∣∣∣∣∣∣∣∣∫ T

0f(s)T dW

∣∣∣∣∣∣∣∣L2(Ω)

= ||f||L2(Ω×[0,T ];Rn) .

Lemma 67.9.2 Let X ≥ 0 and measurable and integrable. Also define a finite measure ν

on B (Rp) by

ν (B)≡∫

Ω

XXB (Y)dP

Then ∫Ω

g(Y)XdP =∫Rp

g(y)dν (y)

where here Y is a measurable function with values in Rp and g ≥ 0 is Borel measurable.Formally, XdP = dν .