Lemma 3.4.2 For d as defined in (3.22) and y7 e C2, we have
|di d+l| = O(At h).
Proof:
di d -= (1 e-(At)) ez (1 -e-1 -+(At)) ( A
v0 i 0 i+1
+ (1- e--i(At) -) (1 e- 7i(At)) --
( i ri+1
= (1 e-'Y(At)) --) -- + (e--At) e-A(At) )
i [i+l + Oi+1
By the mean value theorem, we have
d+1 = (1 e- (t)) Oz z)h + e--(t) ( (At)) h ( +
where (E)) is the derivative of ( ) with respect to z at some zi between z, and
zi+,, =i and 7= are the value and the derivative of 7 at some = between zi and zi+1.
Since Q is bounded and a E C2, both z and A(z) are bounded, and we have
Id d,+I < (1 e-'(Y ) ) ) h + )(At) (z) h
S (At) (z)ih + 5(At) (z)+,h h
= O(At h).