1.3. DERIVATIVE IN THE WHITE NOISE CASE 11

and fn ∈ Cp

∞(Rkn).

The conditional expectation E(Fn|FA) also belongs to S and

is given by

(1.22) E(Fn|FA) = gn(W (h1

n1A),

. . . , W (hkn

n

1A)),

where

gn(x) =

Rkn

fn(y + x)νn(dy),

and νn is the law of the random vector (W (h1

n1Ac

), . . . , W (hkn

n

1Ac )). Furthermore

(1.22) implies

(1.23) Dt(E(Fn|FA)) = E(DFn|FA)1A(t).

We know that E(Fn|FA) converges in

L2

to E(F |FA) = F . Also, (1.23) implies

that D(E(Fn|FA)) converges in

L2(Ω;

H) to E(DF |FA)1A. As a consequence,

using that D is closed, we obtain that

DF = E(DF |FA)1A,

which gives the desired result.