2492 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

Proof: Let t ∈ NCω \D. For t > 0, let t (k) denote the largest point of Pk which is less

than t. Suppose t (m)< t (k). Hence m≤ k. Then

BX (t (m)) = BX0 +∫ t(m)

0Y (s)ds+B

∫ t(m)

0Z (s)dW (s) ,

a similar formula holding for X (t (k)) . Thus for t > t (m) , t /∈ Nω ,

B(X (t)−X (t (m))) =∫ t

t(m)Y (s)ds+B

∫ t

t(m)Z (s)dW (s)

which is the same sort of thing studied so far except that it starts at t (m) rather than at 0and BX0 = 0. Therefore, from Lemma 73.7.1 it follows

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

=∫ t(k)

t(m)

(2⟨Y (s) ,X (s)−X (t (m))⟩+ ⟨BZ,Z⟩L2

)ds

+2∫ t(k)

t(m)

(Z ◦ J−1)∗B(X (s)−X (t (m)))◦ JdW (73.7.37)

Consider that last term. It equals

2∫ t(k)

t(m)

(Z ◦ J−1)∗B

(X (s)−X l

m (s))◦ JdW (73.7.38)

This is dominated by

2∣∣∣∣∫ t(k)

0

(Z ◦ J−1)∗B

(X (s)−X l

m (s))◦ JdW

−∫ t(m)

0

(Z ◦ J−1)∗B

(X (s)−X l

m (s))◦ JdW

∣∣∣∣≤ 4 sup

t∈[0,T ]

∣∣∣∣∫ t

0

(Z ◦ J−1)∗B

(X (s)−X l

m (s))◦ JdW

∣∣∣∣In Lemma 73.6.5 the above expression was shown to converge to 0 in probability. There-fore, by the usual appeal to the Borel Canteli lemma, there is a subsequence still referredto as {m} , such that it converges to 0 pointwise in ω for all ω off some set of measure 0 asm→ ∞. It follows there is a set of measure 0 including the earlier one such that for ω notin that set, 73.7.38 converges to 0 in R. Similar reasoning shows the first term on the rightin the non stochastic integral of 73.7.37 is 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 73.6.2. Finally, it is obvious that

limm→∞

∫ t(k)

t(m)⟨BZ,Z⟩L2

ds = 0 for a.e. ω