71.2. INCLUDING STOCHASTIC INTEGRALS 2429
where M (t) is a local martingale whose quadratic variation satisfies
[M] (t) =∫ t
0∥σ (s,z,ω)−σ (s, ẑ,ω)∥2
L2|u− û|2 ds
Thus, simplifying the constants,
sups∈[0,t]
|u(s)− û(s)|2 ≤C∫ t
0|u(s)− û(s)|2 ds+M∗ (t)
where M∗ (t) = sups∈[0,t] |M (s)|. Then by Gronwall’s inequality,
sups∈[0,t]
|u(s)− û(s)|2 ≤CM∗ (t)
Then take the expectation of both sides. Using the Burkholder Davis Gundy inequality,
E
(sup
s∈[0,t]|u(s)− û(s)|2
)≤CE
((∫ t
0K |z− ẑ|2 |u− û|2 ds
)1/2)
Then adjusting the constant again,
≤ 12
E
(sup
s∈[0,t]|u(s)− û(s)|2
)+CE
(∫ t
0K |z− ẑ|2 ds
)and so,
E
(sup
s∈[0,t]|u(s)− û(s)|2
)≤C
∫ t
0E
(sup
r∈[0,s]|z(s)−ẑ(s)|2
)ds
Letting T z = u where u is defined from z in the integral equation 71.2.7, the above in-equality implies that
E
(sup
s∈[0,t]|T nz1 (s)−T nz2 (s)|2H
)≤C
∫ t
0E
(sup
r∈[0,s]
∣∣T n−1z1 (r)−T n−1z2 (r)∣∣2)ds
≤C2∫ t
0
∫ s
0E
(sup
r1∈[0,r]
∣∣T n−2z1 (r1)−T n−2z2 (r1)∣∣2 )drds
One can iterate this, eventually finding that
E
(sup
s∈[0,t]|T nz1 (s)−T nz2 (s)|2H
)
≤ Cn∫ t
0
∫ t1
0· · ·∫ tn−1
0dtn−1 · · ·dtE
(sup
s∈[0,t]|z1 (s)− z2 (s)|2H
)
=CnT n
(n!)E
(sup
s∈[0,t]|z1 (s)− z2 (s)|2H
)