Une généralisation de l'intégrale stochastique de Wick–Itô

A class of Gaussian processes with fractional spectral measures

“…Our focus is on families of purely singular measures σ with an intrinsic spatial selfsimilarity, typically with Cantor support, and with fractional scaling (and Hausdorff) dimension; see Section 2 below; these are measures with affine selfsimilarity. Note that this notion is different from selfsimilarity in the time-variable; the latter case includes the fractional Brownian motion (fBm), studied in e.g., [2,1,3,14,26]. For the latter (fBm), it is known that the corresponding one-parameter family of measures σ consists of a scale of absolutely continuous measures.…”

“…In the paper [3], see also [1], the case where σ (u) is absolutely continuous with respect to Lebesgue measure, i.e., dσ (u) = m(u) du (where the Radon-Nikodym derivative m satisfies moreover some growth conditions) was considered. The study of [3] included in particular the case of the Brownian motion and of the fractional Brownian motion.…”

“…Refs. [2,1,3,[12][13][14]. In addition, we call attention to the papers [25,26,15,17,29] and the papers cited there.…”

White noise based stochastic calculus associated with a class of Gaussian processes

“…There are quite a number of Gelfand triples associated with W. In [1], [3], and here, we focus on (S 1 , W, S −1 ), namely the Kondratiev space S 1 of stochastic test functions, W defined above, and the Kondratiev space S −1 of stochastic distributions. To define these spaces we first introduce, for k ∈ N, the Hilbert space H k which consists of series of the form (4.4) such that (4.5)…”

An Interpolation Problem for Functions with Values in a Commutative Ring

“…There are quite a number of Gelfand triples associated to L 2 (W). In our previous works [2], [5], and in the present one, we focus on the one consisting of the Kondratiev space S 1 of stochastic test functions, of W, and of the Kondratiev space S −1 of stochastic distributions. To define these spaces we first introduce for k ∈ N the Hilbert space H k which consists of series of the form (1.5) such that (2.10)…”