170
Appendix E
t
0
G
X(t
), t
• dW (t
) = qm − lim
t→0
N
n=0
G (X(t n ), t n ) (t n ),
(E.17)
where
N = (t − t 0 )//t
t n = t 0 + nnt
(t n ) = W (t n+1 ) − W (t n )
(E.18)
The symbol •dW (t ) is used in Eq. E.17 to indicate that is Itô’s integral.
E.1.7.2 Properties of Itô’s Integral
(1)
t
0
W (t
) • dW (t
) =
1
2
W (t)
2
− W (t 0 )
2
− (t − t 0 )
(E.19)
(2)
t
0
W (t
)
n
• dW (t
) =
1
n + 1
W (t)
n+1
− W (t 0 )
n+1
−
n
2
t
0
W (t
)
n−1 dt
(E.20)
where the last integral, on the right, is an usual Riemann’s integral.
(3) The mean over the realizations of any Ito’s integral is null.
t
0
G
X(t
), t
• dW (t
)
= 0.
(E.21)
(4) Itô’s integrals whose integration measures are powers of Wiener’s increment:
t
0
G
X(t
), t
• (dW (t
))
m
≡= qm − lim
t→0
N
n=0
G (X(t n ), t n ) ( (t n ))
m ,
(E.22)
Also can be shown [1–4].
t
0
G
X(t
), t
• (dW (t
))
2
=
t
0
G
X(t
), t
dt
(E.23)
and