Paley–Wiener integral

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.

Definition

Let $i:H\to E$ be an abstract Wiener space with abstract Wiener measure $\gamma$ on $E$ . Let $j:E^{*}\to H$ be the adjoint of $i$ . (We have abused notation slightly: strictly speaking, $j:E^{*}\to H^{*}$ , but since $H$ is a Hilbert space, it is isometrically isomorphic to its dual space $H^{*}$ , by the Riesz representation theorem.)

It can be shown that $j$ is an injective function and has dense image in $H$ .^{[citation needed]} Furthermore, it can be shown that every linear functional $f\in E^{*}$ is also square-integrable: in fact,

\|f\|_{L^{2}(E,\gamma ;\mathbb {R} )}=\|j(f)\|_{H}

This defines a natural linear map from $j(E^{*})$ to $L^{2}(E,\gamma ;\mathbb {R} )$ , under which $j(f)\in j(E^{*})\subseteq H$ goes to the equivalence class $[f]$ of $f$ in $L^{2}(E,\gamma ;\mathbb {R} )$ . This is well-defined since $j$ is injective. This map is an isometry, so it is continuous.

However, since a continuous linear map between Banach spaces such as $H$ and $L^{2}(E,\gamma ;\mathbb {R} )$ is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension $I:H\to L^{2}(E,\gamma ;\mathbb {R} )$ of the above natural map $j(E^{*})\to L^{2}(E,\gamma ;\mathbb {R} )$ to the whole of $H$ .

This isometry $I:H\to L^{2}(E,\gamma ;\mathbb {R} )$ is known as the Paley–Wiener map. $I(h)$ , also denoted $\langle h,x\rangle ^{\sim }$ , is a function on $E$ and is known as the Paley–Wiener integral (with respect to $h\in H$ ).

It is important to note that the Paley–Wiener integral for a particular element $h\in H$ is a function on $E$ . The notation $\langle h,x\rangle ^{\sim }$ does not really denote an inner product (since $h$ and $x$ 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 $\langle h,-\rangle ^{\sim }(x)$ or $I(h)(x)$ rather than using the more compact but potentially confusing $\langle h,x\rangle ^{\sim }$ notation.

References

Park, Chull; Skoug, David (1988), "A Note on Paley-Wiener-Zygmund Stochastic Integrals", Proceedings of the American Mathematical Society, 103 (2): 591–601, doi:10.1090/S0002-9939-1988-0943089-8, JSTOR 2047184
Elworthy, David (2008), MA482 Stochastic Analysis (PDF), Lecture Notes, University of Warwick (Section 6)

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Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

Definition

See also

References