79.2. REPLACING Φ WITH σ (u) 2699
Then by assumption and using Gronwall’s inequality, there is a constant C = C (λ ,K,T )such that
⟨B(u1−u2) ,u1−u2⟩(t)≤CM∗ (t)
Then also, since M∗ is increasing,
sups∈[0,t]
⟨B(u1−u2) ,u1−u2⟩(s)≤CM∗ (t)
Taking expectations and from the Burkholder Davis Gundy inequality,
E
(sup
s∈[0,t]⟨B(u1−u2) ,u1−u2⟩(s)
)
≤ C∫
Ω
(∫ t
0∥σ (w1)−σ (w2)∥2 ⟨B(u1−u2) ,u1−u2⟩
)1/2
dP
≤C∫
Ω
sups∈[0,t]
⟨B(u1−u2) ,u1−u2⟩1/2 (s)(∫ t
0∥σ (w1)−σ (w2)∥2
)1/2
dP
Then it follows after adjusting constants that there exists an inequality of the form
E
(sup
s∈[0,t]⟨B(u1−u2) ,u1−u2⟩(s)
)≤CE
(∫ t
0∥σ (w1)−σ (w2)∥2
L2ds)
Hence
E
(sup
s∈[0,t]⟨B(u1−u2) ,u1−u2⟩(s)
)≤CK2E
(∫ t
0∥w1−w2∥2
W ds)
E
(sup
s∈[0,t]∥u1 (s)−u2 (s)∥2
W
)≤CK2E
(∫ t
0sup
τ∈[0,s]∥w1 (τ)−w2 (τ)∥2
W dτ
)You can iterate this inequality and obtain for ψ (wi) defined as ui in the above, the followinginequality.
E
(sup
s∈[0,t]∥ψnw1−ψ
nw2∥2W (s)
)
≤(CK2)n
E
(∫ t
0
∫ t1
0· · ·∫ tn−1
0sup
tn∈[0,tn−1]
∥w1 (tn)−w2 (tn)∥2W dtn · · ·dt2dt1
)Then, letting t = T,
E
(sup
s∈[0,T ]∥ψnw1−ψ
nw2∥2W (s)
)≤ E
(sup
t∈[0,T ]∥w1 (t)−w2 (t)∥2
)T n
n!
≤ 12
E
(sup
t∈[0,T ]∥w1−w2∥2
W (t)
)