73.4. THE MAIN ESTIMATE 2475
where the convergence of X rk to X in K shows the term 2 ||Y ||K′
∣∣∣∣X rk
∣∣∣∣K is bounded. Thus
the constant C can be assumed to be a continuous function of
||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω)
which equals zero when all are equal to zero and is increasing in each. The term involvingthe stochastic integral is next.
Let M (t) =∫ t
0(Z ◦ J−1
)∗B(X l
k
)τ p ◦ JdW. Then thanks to Corollary 65.11.1
d [M ] =∥∥∥(Z ◦ J−1)∗B
(X l
k
)τ p◦ J∥∥∥2
ds
Applying the Burkholder Davis Gundy inequality, Theorem 63.4.4 for F (r) = r in thatstochastic integral,
2∫
Ω
(sup
t∈[0,T ]
∣∣∣∣∫ t
0
(Z ◦ J−1)∗B
(X l
k
)τ p◦ JdW
∣∣∣∣)
dP
≤C∫
Ω
(∫ T
0
∥∥∥(Z ◦ J−1)∗B(
X lk
)τ p◦ J∥∥∥2
L2(Q1/2U,R)ds)1/2
dP (73.4.13)
So let {gi} be an orthonormal basis for Q1/2U and consider the integrand in the above. Itequals
∞
∑i=1
(((Z ◦ J−1)∗B
(X l
k
)τ p)(J (gi))
)2=
∞
∑i=1
⟨B(
X lk
)τ p,Z (gi)
⟩2
≤∞
∑i=1
⟨B(
X lk
)τ p,(
X lk
)τ p⟩⟨BZ (gi) ,Z (gi)⟩
≤
(sup
tm∈Pk
⟨B(
X lk
)τ p(tm) ,
(X l
k
)τ p(tm)
⟩)∥B∥∥Z∥2
L2
It follows that the integral in 73.4.13 is dominated by
C∫
Ω
suptm∈Pk
⟨B(
X lk
)τ p(tm) ,
(X l
k
)τ p(tm)
⟩1/2∥B∥1/2
(∫ T
0∥Z∥2
L2ds)1/2
dP
Now return to 73.4.12. From what was just shown,
E
(sup
tm∈Pk
⟨B(
X lk
)τ p(tm) ,
(X l
k
)τ p(tm)
⟩)
≤ C+E (|e(k)|)+2∫
Ω
(sup
t∈[0,T ]
∣∣∣∣∫ t
0
(Z ◦ J−1)∗B
(X l
k
)τ p◦ JdW
∣∣∣∣)
dP
≤ C+C∫
Ω
suptm∈Pk
⟨B(
X lk
)τ p(tm) ,
(X l
k
)τ p(tm)
⟩1/2·
∥B∥1/2(∫ T
0∥Z∥2
L2ds)1/2
dP+E (|e(k)|)