73.7. THE ITO FORMULA 2491

Since the sets [τ p = ∞] \ [τ p−1 < ∞] are disjoint, the sum of their probabilities is finite.Hence there is a dominating function in 73.7.32 and so, by the dominated convergencetheorem applied to the sum,

limk→∞

P(Ak) =∞

∑p=0

limk→∞

P(Ak ∩ ([τ p = ∞]\ [τ p−1 < ∞])) = 0

Thus∫ t

t1

⟨Y (s) ,Pn

(Mr

k (s)−Mlk (s)

)⟩ds converges to 0 in probability as k→ ∞.

Now consider∣∣∣∣∫ t

t1

⟨Y (s) ,X r

k (s)−X lk (s)

⟩ds∣∣∣∣ ≤ ∫ T

0|⟨Y (s) ,X r

k (s)−X (s)⟩|ds

+∫ T

0

∣∣∣⟨Y (s) ,X lk (s)−X (s)

⟩∣∣∣ds

≤ 2∥Y (·,ω)∥Lp′ (0,T ) 2−k

for all k large enough, this by Lemma 73.6.2. Therefore,

qk−1

∑j=1

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

⟩converges to 0 in probability. This establishes the desired formula for t ∈ D.

In fact, the formula 73.7.24 is valid for all t ∈ NCω .

Theorem 73.7.2 In Situation 73.2.1, for ω off a set of measure zero, for every t /∈ Nω ,

⟨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 (73.7.34)

Also, there exists a unique continuous, progressively measurable function ⟨BX ,X⟩ such thatit equals ⟨BX (t) ,X (t)⟩ for a.e. t and ⟨BX ,X⟩(t) equals the right side of the above for allt. In addition to this,

E (⟨BX ,X⟩(t)) =

E (⟨BX0,X0⟩)+E(∫ t

0

(2⟨Y (s) ,X (s)⟩+ ⟨BZ,Z⟩L2

)ds)

(73.7.35)

Also the quadratic variation of the stochastic integral in 73.7.34 is dominated by

C∫ t

0∥Z∥2

L2∥BX∥2

W ′ ds (73.7.36)

for a suitable constant C. Also t→ BX (t) is continuous with values in W ′ for t ∈ NCω .