ON STOCHASTIC INTEGRALS ON THE EXTENDED FOCK SPACE AND ITS RIGGINGS

Let µ = µα,β be the so-called generalized Meixner measure ([1, 2]) on the Schwartz distributions space D ′ = D′(R+) (subject to parametrs α and β, µ can be, in particular, the Gaussian, Poissonian, Gamma, Pascal or Meixner mea-sure). Denote by L2(D′, µ) the space of complex-valued square integrable with respect to µ functions on D′. One can show ([1]) that L2(D′, µ) can be identified with the so-called extended Fock space Γext = n=0 H(n)extn!, here the Hilbert spaces H(n)ext depend on the product αβ and consist of (equivalence classes generated by) complex-valued symmetric functions (see the exact definition in [1, 2]). Denote by I: Γext → L2(D′, µ) the corresponding (generalized Wiener-Itô-Sigal) isometrical isomorphism. Note that Γext can be considered as an extension of the symmetric Fock space n=0 L2(R+)⊗̂nn!, here ⊗ ̂ denotes a symmetric tensor power. On the probability space (D′, C(D′), µ), where C(D′) is the generated by cylinder sets σ-algebra on D′, we consider a Meixner random process M · = I(0, 1[0,·), 0,...). It follows from results of [1] thatM is a locally square integrable normal martingale with orthogonal independent increments with respect to the flow of (full and right-continuous by definition) σ-algebras Ft: = σ{Mu: u ≤ t}. We define the Ito ̂ integral on the extended Fock space as the I−1-image of the classical Ito ̂ stochastic integral with respect to the Meixner process. More exactly, we say that a function f ∈ Γext⊗L2(R+) is Ito ̂ integrable if (I ⊗ 1)f ∈ L2(D′, µ)⊗ L2(R+) is adapted with respect to the flow {Ft}t≥0 of generaled by M σ-algebras. In this case we define the Ito ̂ integral I(f) ∈ Γext by the formula I(f): = I−