Operators of Stochastic Differentiation on Spaces of Regular Test and Generalized Functions in the Lévy White Noise Analysis

Марія Миколаївна Дирів, Микола Олександрович Качановський


The operators of stochastic differentiation, which are closely related with stochastic integrals and with the Hida stochastic derivative, play an important role in the classical white noise analysis. In particular, one can use these  operators in order to study properties of solutions of normally ordered stochastic equations, and properties of the  extended Skorohod stochastic integral. So, it is natural to introduce and to study analogs of the mentioned operators in the Lévy white noise analysis. In this paper, using the theory of Hilbert equipments, in terms of the Lytvynov’s generalization of the chaotic representation property we introduce operators of stochastic differentiation on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Lévy white noise functions. Then we establish some properties of introduced operators. This gives a possibility to extend to the Lévy white noise analysis and to deepen the well-known results of the classical white noise analysis that are connected with the operators of stochastic differentiation.


Operator of stochastic differentiation; Extended stochastic integral; Hida stochastic derivative; Lévy process

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J. Bertoin, Lévy Processes.Cambridge:CambridgeUniversity Press, 1996, p. X+265.

P.A. Meyer, “Quantum Probability for Probabilists”, Lect. Notes in Math., vol. 1538, p. X+287, 1993.

Yu.M. Kabanov and A.V. Skorohod, “Extended stochastic integrals”, Proc. School-Symposium Theory Stoch. Proc. Vilnus: Inst. Phys. Math., 1975, pp. 123–167.

D. Surgailis, “On and non-multiple stochastic integration”, Lect. Notes in Control and Inform. Sci., vol. 36, pp. 212–226, 1981.

E.W. Lytvynov, “Orthogonal decompositions for Lévy processes with an application to the gamma, Pascal, and Meixner processes”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., vol. 6, no. 1, pp. 73–102, 2003.

F.E. Benth et al., “Explicit representation of the minimal variance portfolio in markets driven by Lévy processes”, Math. Finance, vol. 13, no. 1, pp. 55–72, 2003.

J.L. Solé et al., “Chaos expansions and Malliavin calculus for Lévy processes”, Stoch. Anal. and Appl., Abel Symposium 2,Berlin: Springer, 2007, pp. 595–612.

N.A. Kachanovsky, “On extended stochastic integrals with respect to Lévy processes”, Carpatian Math. Publ., vol. 5, no. 2, pp. 256–278, 2013.

N.A. Kachanovsky, “Stochastic integral and stochastic de­rivative connected with a Lévy process”, Research Bulletin of NTUU “KPI”, no. 4, pp. 77–81, 2012.

N.A. Kachanovsky, “Extended stochastic integrals with respect to a Lévy processes on spaces of generalized functions”, Math. Bulletin of Taras Shevchenko Sci. Society, no. 10, pp. 169–188, 2013.

M.M. Dyriv and N.A. Kachanovsky, “Stochastic integrals with respect to a Lévy processes and stochastic derivatives on spaces of regular test and generalized functions”, Research Bulletin of NTUU “KPI”, no. 4, pp. 27–30, 2013.

A.S. Ustunel, “An Introduction to Analysis on Wiener Space”, Lecture Notes in Math., vol. 1610, p. 102, 1995.

F.E. Benth, “The Gross derivative of generalized random variables”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., vol. 2, no. 3, pp. 381–396, 1999.

N.A. Kachanovsky, “A generalized Malliavin derivative connected with the Poisson- and Gamma-measures”, Me­thods Funct. Anal. Topol., vol. 9, no. 3, pp. 213–240, 2003.

N.A. Kachanovsky, “A generalized stochastic derivative on the Kondratiev-type space of regular generalized functions of Gamma white noise”, Ibid, vol. 12, no. 4, pp. 363–383, 2006.

N.A. Kachanovsky, “Generalized stochastic derivatives on a space of regular generalized functions of Meixner white noise”, Ibid, vol. 14, no. 1, pp. 32–53, 2008.

N.A. Kachanovsky, “Generalized stochastic derivatives on parametrized spaces of regular generalized functions of Meixner white noise”, Ibid, vol. 14, no. 4, pp. 334–350, 2008.

Yu.M. Berezansky et al., Functional analysis. Birkhauser, 1996, vol. 2, P. 312.

G. Di Nunno et al, “White noise analysis for Levy processes”, J. Funct. Anal., vol. 206, no. 1, pp. 109–148, 2004.

N.A. Kachanovsky, “An extended stochastic integral and the Wick calculus on the connected with the Gamma-measure spaces of regular generalized functions”, Ukr. Math. J., vol. 57, no. 8, pp. 1030–1057, 2005.

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DOI: https://doi.org/10.20535/1810-0546.2014.4.28294


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