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),