2572 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
where Ĉ is a constant used in the Burkholder Davis Gundy inequality. This is a restrictionon the size of K. Thus we only give a solution if K is small enough. Later, this will beremoved in the most interesting case. This will give a local solution valid for a fixed T > 0and then the global solution can be obtained by applying this result on the succession ofintervals [0,T ] , [T,2T ] , [3T,4T ] , and so forth.
From Theorem 76.4.9, if w ∈W , there exists a unique solution u to
Bu(t,ω)−Bu0 (ω)+∫ t
0A(s,u(s,ω) ,ω)ds =
∫ t
0f ds+B
∫ t
0σ (w)dW.
holding in the sense described there. Let ui result from wi. Then from the implicit Itoformula and the above monotonicity estimate,
12⟨B(u1−u2) ,u1−u2⟩(t)+ r
∫ t
0∥u1−u2∥2
W ds
−λ
∫ t
0⟨B(u1−u2) ,u1−u2⟩ds
−∫ 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 (76.5.56)
Therefore, since M∗ is increasing in t, it follows from the Lipschitz condition on σ that
12⟨B(u1−u2) ,u1−u2⟩(t)+ r
∫ t
0∥u1−u2∥2
W ds
−λ
∫ t
0⟨B(u1−u2) ,u1−u2⟩ds−∥B∥K2
∫ t
0∥u1−u2∥2
W ds≤M∗ (t)
Thus, from the assumption about r,
sups∈[0,t]
⟨B(u1−u2) ,u1−u2⟩(s)+4∫ t
0∥u1−u2∥2
W ds
≤ λ
∫ t
0⟨B(u1−u2) ,u1−u2⟩ds+2M∗ (t)
Then applying Gronwall’s inequality,
sups∈[0,t]
⟨B(u1−u2) ,u1−u2⟩(s)+4∫ t
0∥u1−u2∥2
W ds≤ 2eλT M∗ (t)
Now take expectations and use the Burkholder Davis Gundy inequality. The expectation ofthe right side is then dominated by
2ĈeλT∫
Ω
(∫ t
0∥σ (w1)−σ (w2)∥2
L2⟨B(u1−u2) ,u1−u2⟩ds
)1/2
dP