74.6. THE ITO FORMULA 2515

−qk−1

∑j=1

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

⟩(74.6.21)

where tqk = t, ∆X (t j) = X(t j+1

)−X (t j) and e(k)→ 0 in probability. By Lemma 74.5.2

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

2∫ t

0⟨BX ,dM⟩

for ω off a set of measure zero. The deterministic integral on the right converges uniformlyfor t ∈ [0,T ] to

2∫ t

0⟨Y (u) ,X (u)⟩du

Thanks to Lemma 74.5.1. ∣∣∣∣∫ t

0⟨Y (u) ,X (u)⟩du−

∫ t

0⟨Y (u) ,X r

k (u)⟩du∣∣∣∣

≤∫ T

0∥Y (u)∥V ′ ∥X (u)−X r

k (u)∥V

≤ ∥Y∥Lp′ ([0,T ])

(2−k)1/p

for all k large enough. Consider the fourth term. It equals

qk−1

∑j=0

(R−1B

(M(t j+1

)−M (t j)

),M(t j+1

)−M (t j)

)W (74.6.22)

where R−1 is the Riesz map from W to W ′. This equals

14

(qk−1

∑j=0

∥∥R−1BM(t j+1

)+M

(t j+1

)−(R−1BM (t j)+M (t j)

)∥∥2

−qk−1

∑j=0

∥∥R−1BM(t j+1

)−M

(t j+1

)−(R−1BM (t j)−M (t j)

)∥∥2)

From Theorem 63.6.4, as k→ ∞, the above converges in probability to (tqk = t)

14([

R−1BM+M](t)−

[R−1BM−M

](t))≡[R−1BM,M

](t)

Also note that from 74.6.22, this term must be nonnegative since it is a limit of nonnegativequantities. This is what was desired.

Thus all the terms in 74.6.21 converge in probability except for the last term which alsomust converge in probability because it equals the sum of terms which do. It remains tofind what this last term converges to. Thus

⟨BX (t) ,X (t)⟩−⟨BX0,X0⟩= 2∫ t

0⟨Y (u) ,X (u)⟩du