1.3. DERIVATIVE IN THE WHITE NOISE CASE 9
Hence,
E Dϕε(F )
2
H

4
ε2
E 1{0≤F
≤ε}
DF
2
H
4E(|F
|−2
DF
2
H
).
Because ϕε(F ) converges in L2 to 1{F
0}
, Lemma 1.13 implies that 1{F
0}
D1,2,
and by Proposition 1.15 this is possible only if {F 0} has probability zero or
one.
1.3. Derivative in the white noise case
Consider the particular case of a white noise {W (A), A B, µ(A) ∞} on the
measurable space (T, B) with intensity µ. In this case (cf. Itˆ o [17]) the multiple
stochastic integral of order n provides an isometry between the space Ls(T 2 n, Bn, µn)
of square integrable symmetric functions of n variables equipped with the norm

n! · and the nth Wiener chaos Hn.
The multiple stochastic integrals are defined as follows. Consider the set En of
elementary functions of the form
(1.18) f(t1, . . . , tn) =
k
i1,...,in=1
ai1···in 1Ai1
×···×Ain
(t1, . . . , tn),
where A1, . . . , Ak are pairwise disjoint sets of fnite measure, and the coefficients
ai1···in vanish if any two of the indices i1, . . . , in are equal. Then we set
In(f) =
k
i1,...,in=1
ai1···in W (Ai1 ) ··· W (Ain ).
The set En is dense in
L2(T n),
and In is a linear map from En into
L2(Ω)
satisfying
In(f) = In(f) and
E(In(f)Im(g)) =
0 if n = m,
n! f, g
L2(T n)
if n = m,
where f denotes the symmetrization of f. As a consequence, In can be extended
to a linear continuous operator from
L2(T n)
into
L2(Ω).
The image of Ls(T
2 n)
by In is the nth Wiener chaos Hn. This is a consequence of the fact that multiple
stochastic integrals of different order are orthogonal, and that In(f) is a polynomial
of degree n in W (Ai1 ),...,W (Ain ) if f has the form (1.18).
In the case of the Brownian motion, for any f Ls([0,
2 ∞)n)
the multiple
stochastic integral In(f) can be expressed as an iterated Itˆ o integral:
In(f) = n!

0
tn
0
···
t2
0
fn(t1, . . . , tn)dWt1 ··· dWtn .
The derivative DF is a random element in L2(Ω; H), which is isometric to
the space L2(T × Ω, B F, µ × P ). As a consequence, the derivative DF can be
regarded as a stochastic process parameterized by T , that we denote by {DtF, t
T }. Suppose that F is a square integrable random variable having an orthogonal
Wiener chaos expansion of the form
(1.19) F = E(F ) +

n=1
In(fn),
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