6 1. THE DERIVATIVE OPERATOR
The following result characterizes the domain of the derivative operator
D1,2
in terms of the Wiener chaos expansion.
Proposition 1.12. Let F be a square integrable random variable with the
Wiener chaos expansion F =


n=0
JnF . Then F
D1,2
if and only if
(1.12) E( DF
2
H
) =

n=1
n JnF
2
2
∞.
Moreover, if (1.12) holds, then for all n 1 and h H we have
(1.13) D(JnF ), h
H
= Jn−1( DF, h
H
).
Proof. The derivative of a random variable of the form Φa, defined in (1.6),
can be computed using (1.4):
D(Φa) =

a!

j=1

i=1,i=j
Hai (W (ei))Haj
−1
(W (ej ))ej .
Then, using Lemma 1.3, if |a| = n
(1.14) E D(Φa)
2
H
=

j=1
a!

i=1,i=j
ai!(aj 1)!
= |a|,
which implies (1.12) for F = Φa, because JnΦa = Φa. In the general case we can
write
JnF =
a∈Λ,|a|=n
E(F Φa)Φa.
For each n 1 the random variable JnF can be approximated in
L2
by the finite
sum
Jnm)F (
=
a∈Λ,|a|=n,σ(a)=m
E(F Φa)Φa,
as m tends to infinity, where σ(a) = max{i : ai = 0}. Then,
D(Jnm)F (
) =
a∈Λ,|a|=n,σ(a)=m
E(F Φa)D(Φa)
converges as m tends to infinity to
a∈Λ,|a|=n
E(F Φa)D(Φa),
because using (1.14) we get
a∈Λ,|a|=n
(E(F
Φa))2
D(Φa)
2
L2(Ω;H)
=
a∈Λ,|a|=n
n(E(F
Φa))2
= n JnF
2
2
∞.
Hence, using the fact that D is closed (Proposition 1.8), we deduce that JnF belongs
to
D1,2
and
D(JnF ) =
a∈Λ,|a|=n
E(F Φa)D(Φa).
Then, F can be approximated in
L2
by the sequence of random variables F
(N)
=
∑N
n=0
JnF
D1,2,
as N tends to infinity. The sequence of derivatives
DF
(N)
=
a∈Λ,|a|≤N
E(F Φa)D(Φa)
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