In
mathematics, the Paley–Wiener integral is a simple
stochastic integral
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
. When applied to
classical Wiener space
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually ''n''-dimensional Euclidean space). Classical Wiener space i ...
, 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
Raymond Edward Alan Christopher Paley (7 January 1907 – 7 April 1933) was an English mathematician who made significant contributions to mathematical analysis before dying young in a skiing accident.
Life
Paley was born in Bournemouth, Engl ...
and
Norbert Wiener.
Definition
Let
be an
abstract Wiener space The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Camer ...
with abstract Wiener measure
on
. Let
be the
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
of
. (We have abused notation slightly: strictly speaking,
, but since
is a
Hilbert space, it is
isometrically isomorphic to its
dual space , by the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, ...
.)
It can be shown that
is an
injective function and has
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
image in
. Furthermore, it can be shown that every
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
is also
square-integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
: in fact,
:
This defines a natural
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
from
to
, under which
goes to the
equivalence class