10 1. THE DERIVATIVE OPERATOR
where the kernels fn are symmetric functions from
L2(T n).
The derivative DtF
can be easily computed using this expression.
Proposition 1.17. Let F
D1,2
be a random variable with a development of
the form (1.19). Then, we have
(1.20) DtF =

n=1
nIn−1(fn(·, t)).
Proof. Suppose that F = In(fn), where fn is a symmetric and elementary
function of the form (1.18). Then
DtF =
n
j=1
k
i1,...,in=1
ai1···in W (Ai1 ) ··· 1Aij (t) ··· W (Ain ) = nIn−1(fn(·, t)).
The result follow easily.
Suppose that F is a random variable in the space Dk,2 with a Wiener chaos
expansion F = E(F ) +
∑∞
n=1
In(fn). Then, applying Proposition 1.17 k times we
obtain that
DkF
is a random field parametrized by T
k
given by
Dt1,...,tk
k
F =

n=k
n(n 1) ··· (n k + 1)In−k(fn(· , t1, . . . , tk)).
Notice that the L2 norm of this expression is given by (1.15). As a consequence, if
F belongs to D∞,2 = ∩kDk,2, then
(1.21) fn =
1
n!
E(DnF
)
for every n 0 (cf. Stroock [48]).
Stroock’s formula (1.21) is a useful tool to compute Wiener chaos expansions.
For example, suppose that W = {Wt, t 0} is a Brownian motion and F = W1
3.
Then,
f1(t1) = E(Dt1 W1
3)
= 3E(W1
2)1[0,1](t1)
= 31[0,1](t1),
f2(t1, t2) =
1
2
E(Dt1,t2
2
W1
3
) = 3E(W1)1[0,1](t1 t2) = 0,
f3(t1, t2, t3) =
1
6
E(Dt1,t2,t3
3
W1
3
) = 1[0,1](t1 t2 t3),
and we obtain the Wiener chaos expansion
W1
3
= 3W1 + 6
1
0
t1
0
t2
0
dWt1 dWt2 dWt3 .
Let A B. We will denote by FA the σ-field (completed with respect to the
probability P ) generated by the random variables {W (B), B A, µ(B) ∞}. We
will make use of the following technical result.
Proposition 1.18. Let A B and suppose that F D1,2 is FA-measurable.
Then DtF is zero almost everywhere in Ac × Ω.
Proof. Suppose that Fn S converges to F in the norm of the space
D1,2,
where
Fn = fn(W (h1
n
), . . . , W (hkn
n
),
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