2514 CHAPTER 74. A MORE ATTRACTIVE VERSION

and taking limits using the dominated convergence theorem on the sum on the right,

limk→∞

P(Ak) =∞

∑p=1

limk→∞

P(Ak ∩ ([τ p = ∞]\ [τ p−1 ̸= ∞])) = 0

This proves convergence in probability.

limk→∞

P(

supt

∣∣∣∣∫ t

0

⟨B(

X lk

)−BX ,dM

⟩∣∣∣∣> ε

)= 0

Then selecting a subsequence, still denoted with k, we can obtain

P(

supt

∣∣∣∣∫ t

0

⟨B(

X lk

)−BX ,dM

⟩∣∣∣∣> 1k

)< 2−k

and so, by the Borel Cantelli lemma, there is a set of measure zero N such that for thissubsequence, for all ω /∈ N,

supt

∣∣∣∣∫ t

0

⟨B(

X lk

)−BX ,dM

⟩∣∣∣∣≤ 1k

for all k large enough. Thus convergence is uniform.From now on, include N in the exceptional set and every subsequence will be a subse-

quence of this one.

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

Lemma 74.6.1 In Situation 74.1.1, 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

02⟨Y (s) ,X (s)⟩ds

+[R−1BM,M

](t)+2

∫ t

0⟨BX ,dM⟩ (74.6.20)

for R the Riesz map from W to W ′. The covariation term[R−1BM,M

](t) is nonnegative.

Proof: Let t ∈ D. Then t ∈Pk for all k large enough. Consider 74.4.17,

⟨BX (t) ,X (t)⟩−⟨BX0,X0⟩= e(k)+2∫ t

0⟨Y (u) ,X r

k (u)⟩du

+2∫ t

0

⟨BX l

k ,dM⟩+

qk−1

∑j=0

⟨B(M(t j+1

)−M (t j)

),M(t j+1

)−M (t j)

⟩