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.
Let i : H → E be an abstract Wiener space with abstract Wiener measure γ on E. Let j : E∗ → H be the adjoint of i. (We have abused notation slightly: strictly speaking, j : E∗ → H∗, but since H is a Hilbert space, it is isometrically isomorphic to its dual space H∗, by the Riesz representation theorem.)
This defines a natural linear map from j(E∗) to L2(E, γ; R), under which j(f) ∈ j(E∗) ⊆ H goes to the equivalence class [f] of f in L2(E, γ; 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 L2(E, γ; R) is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension I : H → L2(E, γ; R) of the above natural map j(E∗) → L2(E, γ; R) to the whole of H.
This isometry I : H → L2(E, γ; R) is known as the Paley–Wiener map. I(h), also denoted <h, −>∼, is a function on E and is known as the Paley–Wiener integral (with respect to h ∈ H).
It is important to note that the Paley–Wiener integral for a particular element h ∈ H is a function on E. The notation <h, x>∼ 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 prefer to write <h, −>∼(x) or I(h)(x) rather than using the more compact but potentially confusing <h, x>∼ notation.
Other stochastic integrals:
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