67.3. A SPECIAL CASE 2293
Hence the above is dominated by
≤ 14
M2mn−1
∑k=0
E ||∆W (tk)||4U1+
14
M2mn−1
∑k=0
(tk+1− tk)2
≤ M2
4
(mn−1
∑k=0
(C4 +1)(tk+1− tk)2
)
which converges to 0 as n→ ∞. Then from 67.3.8, and referring to 67.3.5,
limn→∞
12
mn−1
∑k=0
(FXX (tk,X (tk))(X (tk+1)−X (tk)) ,(X (tk+1)−X (tk)))H (67.3.9)
= limn→∞
12
mn−1
∑k=0
(FXX (tk,X (tk))
∫ tk+1
tkΦ0dW,
∫ tk+1
tkΦ0dW
)H
= limn→∞
12
mn−1
∑k=0
(FXX (tk,X (tk))Φ0,Φ0)L2(tk+1− tk)
if this last limit exists in L2 (Ω). However, since FXX is bounded, this limit certainly existsfor a.e. ω and equals
=12
∫ T
0(FXX (t,X (t))Φ0,Φ0)L2
dt,
The limit also exists in L2 (Ω) obviously, since FXX is assumed bounded. Therefore, asubsequence of 67.3.9, still denoted as n must converge for a.e. ω to the above integral asn→ ∞.
Next consider 67.3.4.
mn−1
∑k=0
FX (tk,X (tk))(X (tk+1)−X (tk)) =mn−1
∑k=0
FX (tk,X (tk))(∫ tk+1
tkφ 0ds
)
+mn−1
∑k=0
FX (tk,X (tk))∫ tk+1
tkΦ0dW (67.3.10)
Consider the second of these in 67.3.10. From Corollary 65.5.4, it equals
mn−1
∑k=0
∫ tk+1
tkFX (tk,X (tk))Φ0dW
=∫ T
0
(mn−1
∑k=0
X(tk,tk+1] (t)FX (tk,X (tk))
)Φ0dW
which converges as n→ ∞ to ∫ T
0FX (t,X (t))Φ0dW