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# Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis

### Abstract

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, and on spaces of regular test and generalized functions of the Levy white noise analysis. In this paper we make the next step: introduce and study operators of stochastic differentiation on spaces of test functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using Lytvynov's generalization of the chaotic representation property. This can be considered as a contribution in a further development of the Levy white noise analysis.

Key words: Operator of stochastic differentiation, stochastic derivative, extended stochastic integral, Levy process

### Article Information

 Title Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 4, 336-360 MathSciNet MR3469532 zbMATH 06630278 Milestones Received 06/02/2015; Revised 16/02/2015 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

N. A. Kachanovsky
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine

### Citation Example

N. A. Kachanovsky, Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis, Methods Funct. Anal. Topology 21 (2015), no. 4, 336-360.

### BibTex

@article {MFAT782,
AUTHOR = {Kachanovsky, N. A.},
TITLE = {Operators of stochastic differentiation on spaces of nonregular test functions of Lévy white noise analysis},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {4},
PAGES = {336-360},
ISSN = {1029-3531},
MRNUMBER = {MR3469532},
ZBLNUMBER = {06630278},
URL = {http://mfat.imath.kiev.ua/article/?id=782},
}

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