2308 CHAPTER 67. THE EASY ITO FORMULA

such that fk converges to f in L2 (Ω× [0, t] ;Rn,P) with f being progressively measurable.Then by the Itô isometry and the equation

Gk = E (Gk)+∫ t

0fk (s,ω)T dW

you can pass to the limit as k→ ∞ and obtain

F = E (F)+∫ t

0f(s,ω)T dW

Now E (Gk)→ E (F) . Consider the stochastic integrals. By the maximal estimate, Theo-rem 62.9.4, and the Ito isometry,

P

 sups∈[0,t]

nonnegative submartingale︷ ︸︸ ︷∣∣∣∣∫ s

0fk (·,ω)T dW−

∫ s

0f(·,ω)T dW

∣∣∣∣> δ



<

E(∣∣∣∫ t

0 fk (·,ω)T dW−∫ t

0 f(·,ω)T dW∣∣∣2)

δ2

=E(∫ t

0 ∥fk− f∥2Rn ds

)δ

2

From the above convergence result and an application of the Borel Cantelli lemma, thereis a set of measure zero N and a subsequence, still denoted as fk such that for ω /∈ N, theconvergence of the stochastic integrals for this subsequence is uniform. Thus for ω /∈ N,

F = E (F)+∫ t

0f(s,ω)T dW

This proves the existence part of this theorem.It remains to consider the uniqueness. Suppose then that

F = E (F)+∫ T

0f(t,ω)T dW = E (F)+

∫ T

0f1 (t,ω)T dW.

Then ∫ T

0f(t,ω)T dW =

∫ T

0f1 (t,ω)T dW

and so ∫ T

0

(f(t,ω)T − f1 (t,ω)T

)dW = 0

and by the Itô isometry,

0 =

∣∣∣∣∣∣∣∣∫ T

0

(f(t,ω)T − f1 (t,ω)T

)dW∣∣∣∣∣∣∣∣

L2(Ω)

= ||f− f1||L2(Ω×[0,T ];Rn)