70.3. MEASURABILITY IN FINITE DIMENSIONAL PROBLEMS 2401
= |u0|2 h+∫ t−h
0|u(s)|2 ds≤ |u0|2 h+
∫ t
0|u(s)|2 ds
if t ≥ h and if s < h, this is dominated by
|u0|2 t ≤ |u0|2 h≤ |u0|2 h+∫ t
0|u(s)|2 ds
As to the terms from w,∫ t
0|w(s)|2 ds
≤∫ t
0
∣∣∣∣w0 +∫ s
0u(r)dr
∣∣∣∣2 ds≤∫ t
0
(|w0|+
∣∣∣∣∫ s
0u(r)dr
∣∣∣∣)2
ds
≤∫ t
0
(|w0|2 +2 |w0|
∣∣∣∣∫ s
0u(r)dr
∣∣∣∣+ ∣∣∣∣∫ s
0u(r)dr
∣∣∣∣2)
ds
≤ T |w0|2 +T |w0|2 +∫ t
0
∣∣∣∣∫ s
0u(r)dr
∣∣∣∣2 ds+∫ t
0
∣∣∣∣∫ s
0u(r)dr
∣∣∣∣2 ds
≤ 2T |w0|2 +2∫ t
0
(∫ s
0|u(r)|dr
)2
ds≤ 2T |w0|2 +2∫ t
0s∫ s
0|u(r)|2 drds
≤ 2T |w0|2 +2T∫ t
0
∫ s
0|u(r)|2 drds≤ 2T |w0|2 +2T 2
∫ t
0|u(r)|2 dr
From this, the claimed result follows.
Theorem 70.3.3 Suppose N(t,u,v,w,ω) ∈ Rd for u,v,w ∈ Rd , t ∈ [0,T ] and
(t,u,v,w,ω)→ N(t,u,v,w,ω)
is progressively measurable with respect to a constant filtration Ft = F . Also suppose(t,u,v,w)→N(t,u,v,w,ω) is continuous and satisfies C (·,ω)≥ 0 in L1 ([0,T ]) and someµ > 0:
(N(t,u,v,w,ω) ,u)≥−C (t,ω)−µ
(|u|2 + |v|2 + |w|2
).
Also let f be product measurable and f(·,ω) ∈ L2([0,T ] ;Rd
). Then for h > 0, there exists
a product measurable solution u to the integral equation
u(t,ω)−u0(ω)+∫ t
0N(s,u(s,ω),u(s−h,ω) ,w(s,ω) ,ω)ds =
∫ t
0f(s,ω)ds, (70.3.3)
where u0 has values inRd and is F measurable. Here u(s−h,ω)≡ u0 (ω) for all s−h≤ 0and for w0 a given F measurable function,
w(t,ω)≡ w0 (ω)+∫ t
0u(s,ω)ds