2576 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
−∫ t
0⟨Bσ (u1)−Bσ (u2) ,σ (u1)−σ (u2)⟩L2
ds≤M∗ (t)
where the right side is of the form sups∈[0,t] |M (s)| where M (t) is a local martingale havingquadratic variation dominated by
C∫ t
0∥σ (w1)−σ (w2)∥2 ⟨B(u1−u2) ,u1−u2⟩ds (76.5.59)
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⟩(t)
)≤CK2E
(∫ t
0∥w1−w2∥2
W ds)
Thus, for each t ≤ T∫Ω
⟨B(u1−u2) ,u1−u2⟩(t)dP≤CK2E(∫ t
0∥w1−w2∥2
W ds)
one can consider the map ψ (w)≡ u as described above. Then the above inequality implies
E (⟨B(ψnw1−ψnw2) ,ψ
nw1−ψnw2⟩(t))
≤ CK2E(∫ t
0
∥∥ψn−1w1−ψ
n−1w2∥∥2
W dt1
)