2486 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

Then by the Borel Cantelli lemma, one can enlarge the set of measure zero such that forω /∈ N,

supt∈[0,T ]

∣∣∣∣∫ t

0

(Z ◦ J−1)∗B∆k ◦ JdW

∣∣∣∣< 1k

for all k large enough. That is, the claimed uniform convergence holds.From now on, the sequence will either be this subsequence or a further subsequence.

73.7 The Ito FormulaNow at long last, here is the first version of the Ito formula valid on the partition points.

Lemma 73.7.1 In Situation 73.2.1, let D be as above, the union of all the positive meshpoints for all the Pk. Also assume X0 ∈ L2 (Ω;W ) . Then for ω /∈ N the exceptional set ofmeasure zero in Ω and every t ∈ D,

⟨BX (t) ,X (t)⟩= ⟨BX0,X0⟩+∫ t

0

(2⟨Y (s) ,X (s)⟩+ ⟨BZ,Z⟩L2

)ds

+2∫ t

0

(Z ◦ J−1)∗BX ◦ JdW (73.7.24)

where, in the above formula,

⟨BZ,Z⟩L2≡(R−1BZ,Z

)L2(Q1/2U,W)

for R the Riesz map from W to W ′.

Note first that for {gi} an orthonormal basis for Q1/2 (U) ,(R−1BZ,Z

)L2≡∑

i

(R−1BZ (gi) ,Z (gi)

)W = ∑

i⟨BZ (gi) ,Z (gi)⟩W ′W ≥ 0

Proof: Let t ∈ D. Then t ∈Pk for all k large enough. Consider 73.5.18,

⟨BX (t) ,X (t)⟩−⟨BX0,X0⟩= e(k)+2∫ t

0⟨Y (u) ,X r

k (u)⟩du

+2∫ t

0

(Z ◦ J−1)∗BX l

k ◦ JdW +qk−1

∑j=0

⟨B(M(t j+1

)−M (t j)

),M(t j+1

)−M (t j)

⟩−

qk−1

∑j=1

⟨B(∆X (t j)−∆M (t j)) ,∆X (t j)−∆M (t j)

⟩(73.7.25)

where tqk = t, ∆X (t j) = X(t j+1

)−X (t j) and e(k)→ 0 in probability. By Lemma 73.6.5

the stochastic integral on the right converges uniformly for t ∈ [0,T ] to

2∫ t

0

(Z ◦ J−1)∗BX ◦ JdW