On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis

Abstract

We deal with spaces of regular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to study properties of Wick multiplication and of Wick versions of holomorphic functions, and to describe a relationship between Wick multiplication and integration, on these spaces. More exactly, we establish that a Wick product of regular test functions is a regular test function; under some conditions a Wick version of a holomorphic function with an argument from the space of regular test functions is a regular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral with respect to a Lévy process; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the extended stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise. As an example of an application of our results, we consider an integral stochastic equation with Wick multiplication.

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

“…satisfies the Leibnitz rule) with respect to the Wick multiplication. On the above-mentioned spaces of nonregular generalized functions in the Lévy analysis such results were obtained in [26,29], on the spaces of regular generalized functions -in [12,13], on the spaces of regular test functions -in [25].…”

“…Recall that the Wick multiplication on the spaces of regular generalized functions in the Lévy white noise analysis is introduced and studied in [12] (see also [13]), on the spaces of regular test functions -in [25], on the spaces of nonregular generalized functions -in [26] (see also [29]). Now our goal is to introduce and study the Wick multiplication on the spaces of nonregular test functions.…”

“…Finally, we note that by analogy with the regular case [25] (see also [21]) one can introduce so-called Wick versions of holomorphic functions on (H τ ). Namely, let f ∈ (H τ ) and h : C → C be a holomorphic at ( S f )(0) function.…”

“…As in the Gaussian analysis, it is possible to use Lytvynov's generalization of CRP, in particular, in order to construct and study spaces of regular and nonregular test and generalized functions [20], introduce and investigate various operators and operations on these spaces etc. It should be noted that the extended stochastic integral and the Hida stochastic derivative on the spaces of regular test and generalized functions are introduced and studied in [11,20], operators of stochastic differentiation -in [9,10,14], elements of a Wick calculus and its relationship with operators of stochastic differentiation and integration on the spaces of regular generalized functions -in [12,13], on the spaces of regular test functions -in [25]. As for the spaces of nonregular test and generalized functions, the corresponding results are presented in [20,[26][27][28][29].…”

“…The aim of the present paper is to introduce and to study by analogy with [21,25] the Wick product on the spaces of nonregular test functions in the Lévy analysis; to transfer some results of [13,25] to these spaces; and to consider certain related topics, in particular, the relationship between Wick multiplication and stochastic differentiation.…”