Операторы стохастического дифференцирования на пространствах регулярных основных и обобщенных функций в анализе белого шума Леви

Операторы стохастического дифференцирования, тесно свя¬занные со стохастическими интегралами и стохастической производной Хиды, играют важную роль в классическом анализе белого шума. В частности, эти операторы можно использовать для изучения свойств решений нормально упорядоченных стохастических уравнений и свойств расширенного стохастического интеграла Скорохода. Таким образом, естественно вводить и изучать аналоги упомянутых операторов в анализе белого шума Леви. В этой статье, используя теорию гильбертовых оснащений, в терминах литвиновского обобщения свойства хаотического разложения мы вводим операторы стохастического дифференцирования на пространствах параметризованного регулярного оснащения пространства квадратично интегрируемых по мере белого шума Леви функций. Затем мы устанавливаем некоторые свойства введенных операторов. Это дает возможность расширить на анализ белого шума Леви и углубить хорошо известные результаты классического анализа белого шума, связанные с операторами стохастического дифференцирования.

Год издания: 
2014
Номер: 
4
УДК: 
517.98
С. 36–40., Бібліогр.: 20 назв.
Литература: 

1. J. Bertoin, Levy Processes. Cambridge: Cambridge University Press, 1996, p. X+265.
2. P.A. Meyer, “Quantum Probability for Probabilists”, Lect. Notes in Math., vol. 1538, p. X+287, 1993.
3. Yu.M. Kabanov and A.V. Skorohod, “Extended stochastic integrals”, Proc. School-Symposium Theory Stoch. Proc. Vilnus: Inst. Phys. Math., 1975, pp. 123—167.
4. D. Surgailis, “On 2 L and non- 2 L multiple stochastic integration”, Lect. Notes in Control and Inform. Sci., vol. 36, pp. 212—226, 1981.
5. E.W. Lytvynov, “Orthogonal decompositions for Levy 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.
6. F.E. Benth et al., “Explicit representation of the minimal variance portfolio in markets driven by Levy processes”, Math. Finance, vol. 13, no. 1, pp. 55—72, 2003.
7. J.L. Sole et al., “Chaos expansions and Malliavin calculus for Levy processes”, Stoch. Anal. and Appl., Abel Symposium 2, Berlin: Springer, 2007, pp. 595—612.
8. N.A. Kachanovsky, “On extended stochastic integrals with respect to Levy processes”, Carpatian Math. Publ., vol. 5, no. 2, pp. 256—278, 2013.
9. N.A. Kachanovsky, “Stochastic integral and stochastic derivative connected with a Levy process”, Research Bulletin of NTUU “KPI”, no. 4, pp. 77—81, 2012.
10. N.A. Kachanovsky, “Extended stochastic integrals with respect to a Levy processes on spaces of generalized functions”, Math. Bulletin of Taras Shevchenko Sci. Society, no. 10, pp. 169—188, 2013.
11. M.M. Dyriv and N.A. Kachanovsky, “Stochastic integrals with respect to a Levy processes and stochastic derivatives on spaces of regular test and generalized functions”, Research Bulletin of NTUU “KPI”, no. 4, pp. 27—30, 2013.
12. A.S. Ustunel, “An Introduction to Analysis on Wiener Space”, Lecture Notes in Math., vol. 1610, p. 102, 1995.
13. 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. Yu.M. Berezansky et al., Functional analysis. Birkhauser, 1996, vol. 2, P. 312.
19. G. Di Nunno et al, “White noise analysis for Levy processes”, J. Funct. Anal., vol. 206, no. 1, pp. 109—148, 2004.
20. N.A. Kachanovsky, “An extended stochastic integral and the Wick calculus on the connected with the Gammameasure spaces of regular generalized functions”, Ukr. Math. J., vol. 57, no. 8, pp. 1030—1057, 2005.

Транслитерированый список литературы: 

1. J. Bertoin, Lévy Processes. Cambridge: Cambridge University Press, 1996, p. X+265.
2. P.A. Meyer, “Quantum Probability for Probabilists”, Lect. Notes in Math., vol. 1538, p. X+287, 1993.
3. Yu.M. Kabanov and A.V. Skorohod, “Extended stochastic integrals”, Proc. School-Symposium Theory Stoch. Proc. Vilnus: Inst. Phys. Math., 1975, pp. 123–167.
4. D. Surgailis, “On and non- multiple stochastic integration”, Lect. Notes in Control and Inform. Sci., vol. 36, pp. 212–226, 1981.
5. 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.
6. 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.
7. 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.
8. N.A. Kachanovsky, “On extended stochastic integrals with respect to Lévy processes”, Carpatian Math. Publ., vol. 5, no. 2, pp. 256–278, 2013.
9. N.A. Kachanovsky, “Stochastic integral and stochastic derivative connected with a Lévy process”, Research Bulletin of NTUU “KPI”, no. 4, pp. 77–81, 2012.
10. 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.
11. 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.
12. A.S. Ustunel, “An Introduction to Analysis on Wiener Space”, Lecture Notes in Math., vol. 1610, p. 102, 1995.
13. 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.
14. 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.
15. 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.
16. 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.
17. 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.
18. Yu.M. Berezansky et al., Functional analysis. Birkhauser, 1996, vol. 2, P. 312.
19. G. Di Nunno et al, “White noise analysis for Levy processes”, J. Funct. Anal., vol. 206, no. 1, pp. 109–148, 2004.
20. 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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