67.9. SOME REPRESENTATION THEOREMS 2303
Proof: I will show in the process of the proof that functions of the form 67.9.14 are inL2 (Ω,P). Let g ∈ L2 (Ω,Gt ,P) be such that∫
Ω
g(ω)exp(∫ t
0hT dW− 1
2
∫ t
0h ·hdτ
)dP
= exp(−1
2
∫ t
0h ·hdτ
)∫Ω
g(ω)exp(∫ t
0hT dW
)dP = 0
for all such h. It is required to show that whenever this happens for all such functionsexp(∫ t
0 hT dW− 12∫ t
0 h ·hdt)
then g = 0.Letting h be given as above,
∫ t0 hT dW
=m−1
∑i=0
aTi (W(ti+1)−W(ti)) (67.9.15)
=m
∑i=1
aTi−1W(ti)−
m−1
∑i=0
aTi W(ti)
=m−1
∑i=1
(aT
i−1−aTi)
W(ti)+aT0 W(t0)+aT
n−1W(tn) . (67.9.16)
Also 67.9.15 shows exp(∫ t
0 hT dW)
is in L2 (Ω,P) . To see this recall the W(ti+1)−W(ti)are independent and the density of W(ti+1)−W(ti) is
C (n,∆ti)exp
(−1
2|x|2
(ti+1− ti)
), ∆ti ≡ ti+1− ti,
so ∫Ω
(exp(∫ t
0hT dW
))2
dP =∫
Ω
exp(
2∫ t
0hT dW
)dP
=∫
Ω
exp
(m−1
∑i=0
2aTi (W(ti+1)−W(ti))
)dP
=∫
Ω
m−1
∏i=0
exp(2aT
i (W(ti+1)−W(ti)))
dP
=m−1
∏i=0
∫Ω
exp(2aT
i (W(ti+1)−W(ti)))
dP
=m−1
∏i=0
∫Rn
C (n,∆ti)exp(2aT
i x)
exp
(−1
2|x|2
∆ti
)dx < ∞
Choosing the ai appropriately in 67.9.16, the formula in 67.9.16 is of the form
m
∑i=0
yTi Wti