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.be an abstract Wiener space with abstract Wiener measurebe the adjoint of(We have abused notation slightly: strictly speaking,is a Hilbert space, it is isometrically isomorphic to its dual space, by the Riesz representation theorem.)is an injective function and has dense image in[citation needed] Furthermore, it can be shown that every linear functionalis also square-integrable: in fact, This defines a natural linear map fromgoes to the equivalence classThis map is an isometry, so it is continuous.However, since a continuous linear map between Banach spaces such asis uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extensionof the above natural map, also denotedand is known as the Paley–Wiener integral (with respect toIt is important to note that the Paley–Wiener integral for a particular elementdoes not really denote an inner product (sincebelong 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 writerather than using the more compact but potentially confusingOther stochastic integrals:

mathematicsstochastic integralclassical Wiener spaceItō integralRaymond PaleyNorbert Wienerabstract Wiener spaceadjointHilbert spaceisometrically isomorphicdual spaceRiesz representation theoreminjective functionlinear functionalsquare-integrablelinear mapequivalence classisometrycontinuousBanach spacesfunctionabuse of notationCameron–Martin theoremSkorokhod integralStratonovich integralProceedings of the American Mathematical SocietyAnalysistopological vector spacesBochner spaceConvex seriesCylinder set measureInfinite-dimensional vector functionMatrix calculusVector calculusDerivativesDifferentiable vector–valued functions from Euclidean spaceDifferentiation in Fréchet spacesFréchet derivativeFunctional derivativeGateaux derivativeDirectionalGeneralizations of the derivativeHadamard derivativeHolomorphicQuasi-derivativeBesov measureCanonical GaussianClassical Wiener measureMeasureset functionsProjection-valuedVectorBochnerWeaklyStronglymeasurable functionRadonifying functionIntegralsDirect integralDunfordGelfand–Pettis/WeakRegulatedInverse function theoremNash–Moser theoremFeldman–Hájek theoremNo infinite-dimensional Lebesgue measureSazonov's theoremStructure theorem for Gaussian measuresCrinkled arcCovariance operatorFunctional calculusBorel functional calculusContinuous functional calculusHolomorphic functional calculusBanach manifoldbundleConvenient vector spaceChoquet theoryFréchet manifoldHilbert manifoldFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setWeak topologyBanach–Mazur distanceTomita–Takesaki theory