In mathematics, the Paley–Wiener integral is a simple stochastic integral.
When applied to classical Wiener space, it is less general than the Itō integral, but the two agree when they are both defined.
The integral is named after its discoverers, Raymond Paley and Norbert Wiener.
be an abstract Wiener space with abstract Wiener measure
γ
(We have abused notation slightly: strictly speaking,
is a Hilbert space, it is isometrically isomorphic to its dual space
, by the Riesz representation theorem.)
is an injective function and has dense image in
[citation needed] Furthermore, it can be shown that every linear functional
is also square-integrable: in fact, This defines a natural linear map from
, γ ;
goes to the equivalence class
, γ ;
This map is an isometry, so it is continuous.
However, since a continuous linear map between Banach spaces such as
, γ ;
is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension
, γ ;
of the above natural map
, γ ;
, γ ;
, also denoted
and is known as the Paley–Wiener integral (with respect to
It is important to note that the Paley–Wiener integral for a particular element
does not really denote an inner product (since
belong to two different spaces), but is a convenient abuse of notation in view of the Cameron–Martin theorem.
For this reason, many authors[citation needed] prefer to write
rather than using the more compact but potentially confusing
Other stochastic integrals: