2292 CHAPTER 67. THE EASY ITO FORMULA
Consider one of the terms in 67.3.7. Let A ∈Ftk .By Corollary 65.5.4,∫A
12
(FXX (tk,X (tk))
∫ tk+1
tkΦ0dW,
∫ tk+1
tkΦ0dW
)H
dP
=∫
A
12
(∫ tk+1
tkFXX (tk,X (tk))Φ0dW,
∫ tk+1
tkΦ0dW
)H
dP
By independence,
= P(A)12
∫Ω
(∫ tk+1
tkFXX (tk,X (tk))Φ0dW,
∫ tk+1
tkΦ0dW
)H
dP
By the Ito isometry results presented earlier,
=∫
Ω
XA12
∫ tk+1
tk(FXX (tk,X (tk))Φ0,Φ0)L2
dsdP
=∫
A
Ftk measurable︷ ︸︸ ︷12
∫ tk+1
tk(FXX (tk,X (tk))Φ0,Φ0)L2
dsdP
=∫
A
12(FXX (tk,X (tk))Φ0,Φ0)L2
(tk+1− tk)dP
Since A ∈Ftk was arbitrary,
E(
12
(FXX (tk,X (tk))
∫ tk+1
tkΦ0dW,
∫ tk+1
tkΦ0dW
)H|Ftk
)=
12(FXX (tk,X (tk))Φ0,Φ0)L2
(tk+1− tk) .
From what was just shown, and Lemma 67.2.1,
E
([12
mn−1
∑k=0
(FXX (tk,X (tk))Φ0∆W (tk) ,Φ0∆W (tk))H −
mn−1
∑k=0
12(FXX (tk,X (tk))Φ0,Φ0)L2
(tk+1− tk)
]2 (67.3.8)
=14
E
(mn−1
∑k=0
(FXX (tk,X (tk))Φ0∆W (tk) ,Φ0∆W (tk))2H
−mn−1
∑k=0
(FXX (tk,X (tk))Φ0,Φ0)2L2
(tk+1− tk)2
)Now FXX is bounded and so there exists a constant M independent of k and n,
M ≥ ||Φ∗0FXX (tk,X (tk))Φ0|| ,∣∣∣(FXX (tk,X (tk))Φ0,Φ0)L2
∣∣∣