74.6. THE ITO FORMULA 2521

≤ 4 supt∈[0,T ]

∣∣∣∣∫ t

0

⟨B(

X−X lm

),dM

⟩∣∣∣∣In Lemma 74.5.2 the above expression converges to 0. It follows there is a set of measure0 including the earlier one such that for ω not in that set, 74.6.33 converges to 0 in R.Similar reasoning shows the first term on the right in the non stochastic integral of 74.6.32is dominated by an expression of the form

4∫ T

0

∣∣∣⟨Y (s) ,X (s)−X lm (s)

⟩∣∣∣ds

which clearly converges to 0 thanks to Lemma 74.5.1. Finally, it is obvious that

limm,k→∞

[R−1BM,M

](t (k))−

[R−1BM,M

](t (m)) = 0 for a.e. ω

due to the continuity of the quadratic variation.This shows that for ω off a set of measure 0

limm,k→∞

⟨B(X (t (k))−X (t (m))) ,X (t (k))−X (t (m))⟩= 0

Then for x ∈W,

|⟨B(X (t (k))−X (t (m))) ,x⟩|≤ ⟨B(X (t (k))−X (t (m))) ,X (t (k))−X (t (m))⟩1/2 ⟨Bx,x⟩1/2

≤ ⟨B(X (t (k))−X (t (m))) ,X (t (k))−X (t (m))⟩1/2 ∥B∥1/2 ∥x∥W

and solim

m,k→∞

∥BX (t (k))−BX (t (m))∥W ′ = 0

Recall t was arbitrary and {t (k)} is a sequence converging to t. Then the above has shownthat {BX (t (k))}∞

k=1 is a convergent sequence in W ′. Does it converge to BX (t)? Let ξ (t)∈W ′ be what it converges to. Letting v ∈ V then, since the integral equation shows thatt→ BX (t) is continuous into V ′,

⟨ξ (t) ,v⟩= limk→∞

⟨BX (t (k)) ,v⟩= ⟨BX (t) ,v⟩ ,

and now, since V is dense in W, this implies ξ (t) = BX (t) = B(X (t)) since t /∈ Nω . Recallalso that it was shown earlier that BX is weakly continuous into W ′ on [0,T ] hence thestrong convergence of {BX (t (k))}∞

k=1 in W ′ implies that it converges to BX (t), this for anyt ∈ NC

ω .For every t ∈ D and for ω off the exceptional set of measure zero described earlier,

⟨B(X (t)) ,X (t)⟩= ⟨BX0,X0⟩+∫ t

02⟨Y (s) ,X (s)⟩ds+

[R−1BM,M

](t)+2

∫ t

0⟨BX ,dM⟩ (74.6.34)