Wick Analysis for Bernoulli Noise Functionals

Abstract

A Gel’fand tripleS(Ω)⊂L2(Ω)⊂S*(Ω)is constructed of functionals ofZ, whereZ=(Zn)n∈ℕis an appropriate Bernoulli noise on a probability space(Ω,ℱ,ℙ). Characterizations are given to bothS(Ω)andS*(Ω). It is also shown that a Wick-type product can be defined onS*(Ω)and moreoverS*(Ω)forms a commutative algebra with the product. Finally, a transform namedS-transform is defined onS*(Ω)and its continuity as well as other properties are examined.

Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises

“…Lemma 5 (see [4,6]). For ≥ 0, one has { 휎 | ∈ Γ} ⊂ S 푝 ( ) and moreover the system { −푝 휎 휎 | ∈ Γ} forms an orthonormal basis for S 푝 ( ).…”

“…Lemma 6 (see [4,6]). The space S( ) is a nuclear space; namely, for any ≥ 0, there exists > such that the inclusion mapping 푝푞 :…”

“…Definition 9 (see [4,6]). Elements of S * ( ) are called generalized functionals of , while elements of S( ) are called testing functionals of .…”

“…On the other hand, as is shown in [4], one can use the canonical orthonormal basis of L 2 ( ) to construct a nuclear space S( ) such that S( ) is densely contained in L 2 ( ).…”

Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

“…Remark 5.2. In [22], by using the Guichardet representation, the authors defined the Wick product for generalized functionals of Bernoulli noise.…”