73.7. THE ITO FORMULA 2487
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 73.6.2.∣∣∣∣∫ 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
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 (73.7.26)
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))
However, from the description of the quadratic variation of M, the above equals
14
(∫ t
0
∥∥R−1BZ +Z∥∥2
L2ds−
∫ t
0
∥∥R−1BZ−Z∥∥2
L2ds)
which equals ∫ t
0
(R−1BZ,Z
)L2
ds≡∫ t
0⟨BZ,Z⟩L2
ds
This is what was desired.Note that in the case of a Gelfand triple, when W = H = H ′, the term ⟨BZ,Z⟩L2
willend up reducing to nothing more than ∥Z∥2
L2.
Thus all the terms in 73.7.25 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
+2∫ t
0
(Z ◦ J−1)∗BX ◦ JdW +
∫ t
0⟨BZ,Z⟩L2
ds−a