Bounded Operators of Stochastic Differentiation on Spaces of Nonregular Generalized Functions in the Lévy White Noise Analysis

Abstract

Background. Operators of stochastic differentiation play an important role in the Gaussian white noise analysis. In particular, they can be used in order to study properties of the extended stochastic integral and of solutions of normally ordered stochastic equations. Although the Gaussian analysis is a developed theory with numerous applications, in problems of mathematics not only Gaussian random processes arise. In particular, an important role in modern researches belongs to Lévy processes. So, it is necessary to develop a Lévy analysis, including the theory of operators of stochastic differentiation.

Objective. During recent years the operators of stochastic differentiation were introduced and studied, in particular, on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Lévy analysis. In this paper, we make the next step: introduce and study such operators on spaces of nonregular generalized functions.

Methods. We use, in particular, the theory of Hilbert equipments and Lytvynov’s generalization of the chaotic representation property.

Results. The main result is a theorem about properties of operators of stochastic differentiation.

Conclusions. The operators of stochastic differentiation are considered on the spaces of nonregular generalized functions of the Lévy white noise analysis. This can be interpreted as a contribution in a further development of the Lévy analysis. Applications of the introduced operators are quite analogous to the applications of the corresponding ope­rators in the Gaussian analysis.