2588 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
We claim that∫ t
0
⟨(B(
un−∫ (·)
0ΨdW
))′−(
B(
u−∫ (·)
0ΨdW
))′,un−u
⟩ds≥ 0
The difficulty is that∫ (·)
0 ΨdW is only in W . To see that the conclusion is so, note that it isclear from a computation that
∫ t
0
⟨ 1−τ(h)h
(Bun−B
∫ (·)0 ΨdW
)− 1−τ(h)
h
(Bu−B
∫ (·)0 ΨdW
),un−u
⟩ds≥ 0 (76.7.68)
Claim: The above is indeed nonnegative.Proof: Denote by q(t) the stochastic integral, un as u and u as v to save notation. Then
the left side of the above equals
1h
∫ t
0⟨B(u−q)−B(v−q) ,u− v⟩ds
−1h
∫ t
h⟨B(u(s−h)−q(s−h))−B(v(s−h)−q(s−h)) ,u(s)− v(s)⟩ds
≥ 1h
∫ t
0⟨B(u−q)−B(v−q) ,u− v⟩ds
− 12h
∫ t
h
⟨B(u(s−h)−q(s−h))−B(v(s−h)−q(s−h)) ,
(u(s−h)− v(s−h))
⟩ds
− 12h
∫ t
h⟨B(u−q)−B(v−q) ,u− v⟩ds
≥ 1h
∫ t
0⟨B(u−q)−B(v−q) ,u− v⟩ds
− 12h
∫ t−h
0⟨B(u(s)−q(s))−B(v(s)−q(s)) ,(u(s)− v(s))⟩ds
− 12h
∫ t
h⟨B(u−q)−B(v−q) ,u− v⟩ds
=1h
∫ t
t−h⟨B(u−q)−B(v−q) ,u− v⟩ds
+1h
∫ t−h
0⟨B(u−q)−B(v−q) ,u− v⟩ds
− 12h
∫ t−h
0⟨B(u−q)−B(v−q) ,(u− v)⟩ds
− 12h
∫ t
h⟨B(u−q)−B(v−q) ,u− v⟩ds