As in the previous Lectures, we consider a filtered probability space on which is defined a Brownian motion , and we assume that is the usual completion of the natural filtration of . Our goal is here to write an orthogonal decomposition of the space that is particularly suited to the study of the space . For simplicity of the exposition, we restrict ourselves to the case where the Brownian motion is one-dimensional.
In the sequel, for , we denote by the simplex and if ,
is called the space of Wiener chaos of order . By convention the set of constant random variables shall be denoted by .
By using the Itō’s isometry, we readily compute that
As a consequence, the spaces are orthogonal in . It is easily seen that is the closure of the linear span of the family
where for , we denoted by the map such that . It turns out that can be computed by using Hermite polynomials. The Hermite polynomial of order is defined as
By the very definition of , we see that for every ,
Lemma. If then
Proof. On one hand, we have for ,
On the other hand, for , let us consider
From Itō’s formula, we have
By iterating the previous linear relation, we easily obtain that for every ,
As we pointed it out, for , the spaces and are othogonal. We have the following orthogonal decomposition of :
Theorem.[Wiener chaos expansion]
Proof. As a by-product of the previous proof, we easily obtain that for ,
where the convergence of the series is almost sure but also in . Therefore, if is orthogonal to , then is orthogonal to every , . This implies that
As we are going to see, the space or more generally is easy to describe by using the Wiener chaos expansion. The keypoint is the following proposition:
Proposition. Let , then and where for ,
Proof. Let . We have
Thus is a smooth cylindric functional and
It is easy to see that , therefore we have
As a consequence, we compute that We now observe that is the closure in of the linear span of the family
to conclude the proof of the proposition
We can finally turn to the description of using the chaos decomposition:
Theorem. Let and let
be the chaotic decomposition of . Then if and only if
and in that case,
Proof. It is a consequence of the fact that for , .
An immediate but useful corollary of the previous theorem is the following result:
Corollary. Let be a sequence in that converges to in and such that
Exercise. Let and let
be the chaotic decomposition of . Show that that , if and only if
Exercise. Let . Show that for , .