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.

The integral is named after its discoverers, Raymond Paley and Norbert Wiener.

Definition

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Let   be an abstract Wiener space with abstract Wiener measure   on  . Let   be the adjoint 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.)

It can be shown that   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   to  , under which   goes to the equivalence class   of   in  . This is well-defined since   is injective. This map is an isometry, so it is continuous.

However, since a continuous linear map between Banach spaces such as   and   is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension   of the above natural map   to the whole of  .

This isometry   is known as the Paley–Wiener map.  , also denoted  , is a function on   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   is a function on  . The notation   does not really denote an inner product (since   and   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   or   rather than using the more compact but potentially confusing   notation.

See also

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Other stochastic integrals:

References

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  • 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)