74.6. THE ITO FORMULA 2519

This is so because if τ p = ∞, then it has no effect but also it could happen that the defin-ing inequality may hold even if τ p < ∞ hence the inequality. This is no larger than anexpression of the form

Cn

ε

∫Ω

∫ T

0∥Y (s)∥V ′

∥∥∥(Mτ p)rk (s)− (Mτ p)l

k (s)∥∥∥

WdsdP (74.6.28)

The inside integral converges to 0 by continuity of M. Also, thanks to the stopping time,the inside integral is dominated by an expression of the form∫ T

0∥Y (s)∥V ′ 2pds

and this is a function in L1 (Ω) by assumption on Y . It follows that the integral in 74.6.28converges to 0 as k→ ∞ by the dominated convergence theorem. Hence

limk→∞

P(Ak ∩ ([τ p = ∞])) = 0.

Since the sets [τ p = ∞] \ [τ p−1 < ∞] are disjoint, the sum of their probabilities is finite.Hence there is a dominating function in 74.6.27 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)1/p

for all k large enough, this by Lemma 74.5.1. 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 74.6.20 is valid for all t ∈ NCω .

Theorem 74.6.2 In Situation 74.1.1, for ω off a set of measure zero, it follows that forevery t ∈ NC

ω ,

⟨BX (t) ,X (t)⟩= ⟨BX0,X0⟩+∫ t

02⟨Y (s) ,X (s)⟩ds