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.
Highlights
Operators of stochastic differentiation play an important role in the Gaussian white noise analysis
The aim of this paper is to introduce the operators of stochastic differentiation on the spaces of nonregular generalized functions of the Lévy white noise analysis; and to establish some properties of these operators
F (m) are the kernels from decompositions (6) and (3) for F and f respectively, ext denotes the dual pairings between elements of negative and positive spaces from chains (5), these pairings are generated by the scalar products in Following [13], we recall a notion of the extended stochastic integral on (H ) q H
Summary
Operators of stochastic differentiation play an important role in the Gaussian white noise analysis. They can be used in order to study properties of the extended stochastic integral and of solutions of normally ordered stochastic equations. 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. 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.
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More From: Research Bulletin of the National Technical University of Ukraine "Kyiv Polytechnic Institute"
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