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.
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