2452 CHAPTER 72. THE HARD ITO FORMULA

72.5 Converging In ProbabilityI am working toward the Ito formula 72.3.3. In order to get this, there is a technical resultwhich will be needed.

Lemma 72.5.1 Let X (s)−X lk (s)≡ ∆k (s) . Then the following limit occurs.

limk→∞

P

([sup

t∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗∆k (s)

)◦ JdW (s)

∣∣∣∣≥ ε

])= 0. (72.5.11)

That is,

supt∈[0,T ]

∣∣∣∣∫ t

0R((

Z (s)◦ J−1)∗(X (s)−X lk (s)

))◦ JdW (s)

∣∣∣∣converges to 0 in probability. Also the stochastic integral makes sense because X is Hpredictable.

Proof: First note that from Lemma 72.4.1, for a.e. ω, X (t) has values in H for t ∈ [0,T ]and so it makes sense to consider it in the stochastic integral provided it is H progressivelymeasurable. However, as noted in Situation 72.3.1, this function is automatically V ′ pre-dictable. Therefore,

⟨X (t) ,v⟩= (X (t) ,v)

is real predictable for every v ∈V. Now if h ∈ H, let vn→ h in H and so for each ω,

(X (t,ω) ,vn)→ (X (t,ω) ,h)

By the Pettis theorem, X is H predictable, hence progressively measurable. Also it wasshown above that t → X (t) is weakly continuous into H. Therefore, the desired resultfollows from Lemma 65.14.3 on Page 2267.

72.6 The Ito FormulaNow at long last, here is the first version of the Ito formula.

Lemma 72.6.1 In Situation 72.3.1, let D be as above, the union of all the positive meshpoints for all the Pk. Also assume X0 ∈ L2 (Ω;H) . Then for every t ∈ D,

|X (t)|2 = |X0|2 +∫ t

0

(2⟨Y (s) , X̄ (s)⟩+ ||Z (s)||2

L2(Q1/2U,H)

)ds

+2∫ t

0R((

Z (s)◦ J−1)∗X (s))◦ JdW (s) (72.6.12)

Note that it was shown above that X (t,ω) has values in H for a.e. ω .

Proof: Let t ∈ D. Then t ∈Pk for all k large enough. Consider 72.4.10,

|X (t)|2−|X0|2 = 2∫ t

0⟨Y (u) ,X r

k (u)⟩du