Paley-Wiener integral

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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.

[edit] Definition

Let $i : H \to E$ be an abstract Wiener space with abstract Wiener measure $γ$ 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 injective and has dense image in $H$ . Furthermore, it can be shown that every linear functional $\ell \in E^{*}$ is also square-integrable: in fact,

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

This defines a natural linear map from $j (E *)$ to $L^{2} (E, \gamma; \mathbb{R})$ , under which $j(\ell) \in j(E^{*}) \subseteq H$ goes to the equivalence class of $\ell$ 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 $j(\ell) \mapsto [\ell]$ 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 $\langle h, - \rangle^{\sim}$ , 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 (after all, $h$ 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 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.