2574 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

and (t,u,ω)→ A(t,u,ω) is progressively measurable. Also suppose for p ≥ 2, the coer-civity, and the boundedness conditions

λ ⟨Bu,u⟩+ ⟨A(t,u,ω) ,u⟩V ≥ δ ∥u∥pV − c(t,ω) (76.5.57)

for all λ large enough.

∥A(t,u,ω)∥V ′ ≤ k∥u∥p−1V + c1/p′ (t,ω) (76.5.58)

where c ∈ L1 ([0,T ]×Ω). Also suppose the monotonicity condition that for all r > 0 thereexists λ such that

⟨(λB+A(ω))(u)− (λB+A(ω))(v) ,u− v⟩ ≥ r∥u− v∥2W

Also suppose that

(t,u,ω) ∈ [0,T ]×W ×Ω→ σ (t,u,ω) is progressively measurable into W

∥σ (t,u,ω)∥W ≤C+C∥u∥W∥σ (t,u,ω)−σ (t, û,ω)∥L2(Q1/2U,W) ≤ K ∥u− û∥W

Then if u0 ∈ L2 (Ω,W ) with u0 F0 measurable, there exists a unique solution u(·,ω) ∈ Vω

with u ∈ V (Lp ([0,T ]×Ω,V ) and progressively measurable) such that for ω off a set ofmeasure zero,

Bu(t,ω)−Bu0 (ω)+∫ t

0A(s,u(s,ω) ,ω)ds =

∫ t

0f ds+B

∫ t

0σ (u)dW.

In case B is the Riesz map, you do not have to make any assumption on the size of K.Thus

⟨Bu,u⟩= ∥u∥2W

The case of most interest is the usual one where V ⊆W =W ′ ⊆ V ′, the case of a Gelfandtriple in which B is the identity. As to σ , the assumption is made that

∥σ (t,ω,u)∥W ≤C+C∥u∥W

∥σ (t,ω,u1)−σ (t,ω,u2)∥L2(Q1/2U,W) ≤ K ∥u1−u2∥WOf course it is also assumed that whenever u has values in W and is progressively measur-able, (t,ω)→ σ (t,ω,u(t,ω)) is also progressively measurable into L2

(Q1/2U,W

).

Letting wi ∈ L2 ([0,T ]×Ω,W ) each wi being progressively measurable, the above as-sumptions imply that there exists a solution ui to the integral equation

Bui (t,ω)−Bu0 (ω)+∫ t

0A(ui)ds =

∫ t

0f (s,ω)ds+B

∫ t

0σ (wi)dW

here we write σ (wi) for short instead of σ (t,ω,wi). First, consider

w ∈ L2 ([0,T ]×Ω,W )∩L∞([0,T ] ,L2 (Ω,W )

)