2226 CHAPTER 64. WIENER PROCESSES

Thus, this has shown that for all A ∈Fs,∫A

E((W (t) ,h)2 |Fs

)dP−

∫A(W (s) ,h)2 dP

= P(A)(t− s)(Qh,h) =∫

A(t− s)(Qh,h)dP

and since A ∈Fs is arbitrary, this proves

E((W (t) ,h)2− t (Qh,h) |Fs

)= (W (s) ,h)2− s(Qh,h)

This proves one half of the theorem.Next suppose both {W (t)} and

{(W (t) ,h)2− t (Qh,h)

}are martingales for any h∈H.

It follows that both {(W (t) ,h)} and{(W (t) ,h)2− t (Qh,h)

}are martingales also. There-

fore, by Levy’s theorem, Theorem 63.8.5, {(W (t) ,h)} is a Wiener process with the prop-erty that its variance at t equals (Qh,h) t instead of t. Thus the time increments are normaland independent. I need to verify that {W (t)} is a Q Wiener process. One of the thingswhich needs to be shown is that

E ((W (t)−W (s) ,h1)(W (t)−W (s) ,h2)) = (Qh1,h2)(t− s) . (64.6.41)

I have just shownE((W (t)−W (s) ,h)2

)= (t− s)(Qh,h) (64.6.42)

which follows from Levy’s theorem which concludes {(W (t) ,h)} is a Wiener process.Therefore,

E ((W (t)−W (s) ,h1 +h2)(W (t)−W (s) ,h2 +h1))

= (Q(h1 +h2) ,(h1 +h2))(t− s)

Now using 64.6.42, it follows from this that

E ((W (t)−W (s) ,h1)(W (t)−W (s) ,h2)) = (Qh1,h2)(t− s)

which shows 64.6.41. This completes the proof.