2484 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE
for all k large enough. By the usual proof of completeness of Lp, it follows that X lnk(t)→
X (t) for a.e. t, this for each ω /∈ N, a similar assertion holding for X rnk
. We denote thesesubsequences as
{X r
k
}∞
k=1 ,{
X lk
}∞
k=1 .Now with this preparation, it is possible to show the desired convergence.
Lemma 73.6.3 In the above context, let X (s)−X lk (s)≡ ∆k (s) . Then the integral∫ t
0
(Z ◦ J−1)∗BX ◦ JdW
exists as a local martingale and the following limit is valid for the subsequence of Lemma73.6.2
limk→∞
P
([sup
t∈[0,T ]
∣∣∣∣∫ t
0
(Z ◦ J−1)∗B∆k ◦ JdW
∣∣∣∣≥ ε
])= 0.
That is,
supt∈[0,T ]
∣∣∣∣∫ t
0
(Z ◦ J−1)∗B∆k ◦ JdW
∣∣∣∣converges to 0 in probability.
Proof: In the argument τm will be defined in 73.6.20. Let
Ak ≡
{ω : sup
t∈[0,T ]
∣∣∣∣∫ t
0
(Z ◦ J−1)∗B∆k ◦ JdW
∣∣∣∣≥ ε
}
then
Ak ∩{ω : τm = ∞} ⊆
{ω : sup
t∈[0,T ]
∣∣∣∣∫ t
0
(Z ◦ J−1)∗B∆
τmk ◦ JdW
∣∣∣∣≥ ε
}By Burkholder Davis Gundy inequality,
P(Ak ∩{ω : τm = ∞}) ≤ Cε
∫Ω
supt∈[0,T ]
∣∣∣∣∫ t
0
(Z ◦ J−1)∗B∆
τmk ◦ JdW
∣∣∣∣dP
≤ Cε
∫Ω
(∫ T
0∥Z∥2
L2
∥∥B∆τmk
∥∥2 dt)1/2
dP
≤ Cε
(∫Ω
∫ T
0∥Z∥2
L2
∥∥B∆τmk
∥∥2 dtdP)1/2
Recall that if ⟨Bx,x⟩ ≤ m, then ∥Bx∥W ′ ≤ m1/2 ∥B∥1/2. Then the integrand is bounded fora.e. t by ∥Z∥2
L24m∥B∥ . Next use the result of Lemma 73.6.2 and the dominated con-
vergence theorem to conclude that the above converges to 0 as k → ∞. Then from theassumption that τm = ∞ for all m large enough,
P(Ak) =∞
∑m=1
P(Ak ∩ ([τm = ∞]\ [τm−1 < ∞]))