Skip to main content
SpringerLink
Log in

A generalization of an extended stochastic integral

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We propose a generalization of an extended stochastic integral to the case of integration with respect to a broad class of random processes. In particular, we obtain conditions for the coincidence of the considered integral with the classical Itô stochastic integral.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Hitsuda, “Formula for Brownian partial derivatives,” in: Proceedings of the Second Japan-USSR Symposium on Probability Theory, Vol. 2 (1972), pp. 111–114.

    Google Scholar 

  2. Ju. L. Daletsky and S. N. Paramonova, “Stochastic integrals with respect to a normally distributed additive set function,” Dokl. Akad. Nauk SSSR, 208,No. 3, 512–515 (1973).

    MathSciNet  Google Scholar 

  3. Ju. L. Daletsky and S. N. Paramonova, “A formula of the theory of Gaussian measures and the estimation of stochastic integrals,” Teor. Ver. Primen., 19,No. 4, 844–849 (1974).

    Google Scholar 

  4. A. V. Skorokhod, “On the generalization of the stochastic integral,” Teor. Ver. Primen., 20,No. 2, 223–228 (1975).

    Google Scholar 

  5. Yu. M. Kabanov and A. V. Skorokhod, “Extended stochastic integrals,” in: Proceedings of the School and Seminar on the Theory of Random Processes [in Russian], Part 1 (1975), pp. 123–167.

  6. Yu. M. Kabanov, “Extended stochastic integrals,” Teor. Ver. Primen., 20,No. 4, 725–737 (1975).

    MathSciNet  Google Scholar 

  7. N. A. Kachanovsky, “On the extended stochastic integral connected with the gamma-measure on an infinite-dimensional space,” Meth. Funct. Anal. Topol., 8,No. 2, 10–32 (2002).

    MATH  MathSciNet  Google Scholar 

  8. N. A. Kachanovsky, “On the extended stochastic integral and Wick calculus on the spaces of regular generalized functions connected with the Gamma-measure,” Ukr. Mat. Zh., 57,No. 8, 1030–1057 (2005).

    MATH  MathSciNet  Google Scholar 

  9. T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise. An Infinite-Dimensional Calculus, Kluwer, Dordrecht (1993).

    MATH  Google Scholar 

  10. N. V. Norin, The Extended Stochastic Integral in Linear Spaces with Differentiable Measures and Related Topics, World Scientific, Singapore (1996).

    MATH  Google Scholar 

  11. Yu. M. Berezansky and V. A. Tesko, “Spaces of fundamental and generalized functions associated with generalized translation,” Ukr. Mat. Zh., 55,No. 12, 1907–1979 (2003).

    Article  Google Scholar 

  12. Yu.M. Berezansky and V. A. Tesko, “Orthogonal approach to the construction of the theory of generalized functions of infinitely many variables and the Poisson or white noise analysis,” Ukr. Mat. Zh., 56,No. 12, 1885–1914 (2004).

    Article  Google Scholar 

  13. S. Albeverio, Yu. G. Kondratiev, and L. Streit, “How to generalize white noise analysis to non-Gaussian measures,” in: P. Blanchard et al. (editors), Dynamics of Complex and Irregular Systems, World Scientific, Singapore (1993), pp. 120–130.

    Google Scholar 

  14. S. Albeverio, Yu. L. Daletsky, Yu. G. Kondratiev, and L. Streit, “Non-Gaussian infinite-dimensional analysis,” J. Funct. Anal., 138,No. 2, 311–350 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  15. E. W. Lytvynov and G. F. Us, “Dual Appell systems in non-Gaussian white noise calculus,” Meth. Funct. Anal. Topol., 2,No. 2, 70–85 (1996).

    MATH  MathSciNet  Google Scholar 

  16. B. Simon, The P(ϕ)2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, NJ (1974).

    Google Scholar 

  17. Yu. M. Berezansky, N. W. Zhernakov, and G. F. Us, “Operator stochastic integrals,” Ukr. Mat. Zh., 39,No. 2, 144–149 (1987).

    Article  Google Scholar 

  18. Yu. M. Berezansky, N. W. Zhernakov, and G. F. Us, “A spectral approach to quantum stochastic integrals,” Rep. Math. Phys., 28,No. 3, 347–360 (1988).

    Article  Google Scholar 

  19. P. A. Meyer, “Quantum probability for probabilists,” in: Lect. Notes Math., Springer, Berlin (1993).

    Google Scholar 

  20. Yu. M. Berezansky, “On the theory of commutative Jacobi fields,” Meth. Funct. Anal. Topol., 4,No. 1, 1–31 (1998).

    MathSciNet  Google Scholar 

  21. Yu. G. Kondratiev, J. L. Da Silva, L. Streit, and G. F. Us, “Analysis on Poisson and Gamma spaces,” Inf. Dimen. Anal. Quant. Probab. Relat. Top., 1,No. 1, 91–117 (1998).

    Article  MATH  Google Scholar 

  22. Yu. G. Kondratiev and E. W. Lytvynov, “Operators of Gamma white noise calculus,” Inf. Dimen. Anal. Quant. Probab. Relat. Top., 3,No. 3, 303–335 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  23. W. Schoutens, Stochastic Processes and Orthogonal Polynomials, Springer, New York (2000).

    MATH  Google Scholar 

  24. Yu. M. Berezansky and D. A. Mierzejewski, “The chaotic decomposition for the Gamma field,” Funct. Anal. Appl., 35, 263–266 (2001).

    Article  Google Scholar 

  25. Yu. M. Berezansky and D. A. Mierzejewski, “The construction of the chaotic representation for the gamma field,” Inf. Dimen. Anal. Quant. Probab. Relat. Top., 6,No. 1, 33–56 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  26. Yu. M. Berezansky, “Pascal measure on generalized functions and the corresponding generalized Meixner polynomials,” Meth. Funct. Anal. Topol., 8,No. 1, 1–13 (2002).

    MathSciNet  Google Scholar 

  27. D. A. Mierzejevski, “Generalized Jacobi fields,” Meth. Funct. Anal. Topol., 9,No. 1, 80–100 (2003).

    Google Scholar 

  28. E. Lytvynov, “Polynomials of Meixner’s type in infinite dimensions—Jacobi fields and orthogonality measures,” J. Funct. Anal., 200, 118–149 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  29. E. Lytvynov, “Orthogonal decompositions for Levy processes with applications to the Gamma, Pascal, and Meixner processes,” Inf. Dimen. Anal. Quant. Probab. Relat. Top., 6, 73–102 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  30. I. Rodionova, “Analysis connected with generating functions of exponential type in one and infinitely many dimensions,” Meth. Funct. Anal. Topol., 11,No. 3, 275–297 (2005).

    MATH  MathSciNet  Google Scholar 

  31. D. Nualart, The Malliavin Calculus and Related Topics, Springer, New York (1995).

    MATH  Google Scholar 

  32. B. Øksendal, “An introduction to Malliavin calculus with applications to economics,” in: Lect. Notes (Norwegian School of Economics and Business Administration), Norway (1997).

  33. F. Flandoli and F. Russo, “Generalized integration and stochastic ODEs,” Ann. Probab., 30,No. 1, 270–292 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  34. B. Rüdiger, “Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces,” Stoch. Rep., 76,No. 3, 213–242 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  35. S. Albeverio and B. Rüdiger, “Stochastic integrals and the Lévy-Itô decomposition theorem on separable Banach spaces,” Stochast. Anal. Appl., 23,No. 2, 217–253 (2005).

    Article  MATH  Google Scholar 

  36. A. Mandrekar and B. Rüdiger, “Levy noises and stochastic integrals on Banach spaces,” in: “Stochastic Partial Differential Equations and Applications VII” (Lect. Notes Pure Appl. Math., 245) (2006).

  37. Yu. M. Berezansky and V. A. Tesko, On a Generalization of an Extended Stochastic Integral (in Ukrainian), Preprint No. 2005.2, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2003).

    Google Scholar 

  38. Yu. M. Berezansky, “Commutative Jacobi fields in Fock space,” Integr. Equat. Operator Theory, 30, 163–190 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  39. Yu. M. Berezansky, “Spectral theory of commutative Jacobi fields: direct and inverse problems,” Fields Inst. Conmun., 25, 211–224 (2000).

    MathSciNet  Google Scholar 

  40. Yu. M. Berezansky and Yu. G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Vols 1, 2, Kluwer, Dordrecht (1995).

    Google Scholar 

  41. Yu. M. Berezansky, “Poisson measure as the spectral measure of Jacobi fields,” Inf. Dimen. Anal. Quant. Probab. Relat. Top., 3,No. 1, 121–139 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  42. M. G. Krein, “Fundamental aspects of the theory of representation of Hermitian operators with deficiency index (m, m),” Ukr. Mat. Zh., 1,No. 2, 3–66 (1949).

    MathSciNet  Google Scholar 

  43. M. G. Krein, “Infinite J-matrices and a matrix moment problem,” Dokl. Akad. Nauk SSSR, 69,No. 2, 125–128 (1949).

    MATH  MathSciNet  Google Scholar 

  44. Yu. M. Berezansky, Z. G. Sheftel, and G. F. Us, Functional Analysis, Birkhauser, Basel (1996).

    MATH  Google Scholar 

  45. E. W. Lytvynov, “Multiple Wiener integrals and non-Gaussian white noise: a field approach,” Meth. Funct. Anal. Topol., 1,No. 1, 61–85 (1995).

    MATH  MathSciNet  Google Scholar 

  46. Y. Itô and I. Kubo, “Calculus on Gaussian and Poissonian white noise,” Nagoya Math. J., 111, 41–84 (1988).

    MATH  MathSciNet  Google Scholar 

  47. N. Privault and J. L. Wu, “Poisson stochastic integration in Hilbert spaces,” Ann. Math. Univ. B. Pascal, 6, 41–61 (1999).

    MATH  MathSciNet  Google Scholar 

  48. N. I. Akhiezer and I. M. Glazman, The Theory of Linear Operators in Hilbert Space, Ungar, New York (1961).

    Google Scholar 

  49. M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Spaces [in Russian], Leningrad University, Leningrad (1980).

    Google Scholar 

  50. R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  51. I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Vol. 3, Nauka, Moscow (1975).

    MATH  Google Scholar 

  52. A. D. Wentzell, Theorie Zufälliger Prozesse, Akademie, Berlin (1979).

    Google Scholar 

  53. P. E. Protter, Stochastic Integration and Differential Equations, 2nd ed., Springer, Berlin (2004).

    MATH  Google Scholar 

  54. C. Dellacherie, B. Maisonneuve, and P. A. Meyer, “Probabilites et Potentiel. Ch. XVII a XXIV. Processus de Markov (fin),” in: Complem. Calcul. Stochast., Hermann, Paris (1992).

    Google Scholar 

  55. S. Albeverio and N. Smorodina, The Poisson Analog of the Multiple Wiener-Itô Stochastic Integral, Preprint No. 21, Bonn (2002).

  56. S. Albeverio and N. Smorodina, The Multiple Stochastic Integrals and the Transformations of the Poisson Measure, Preprint No. 212, Bonn (2005).

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany

    S. Albeverio

  2. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv

    Yu. M. Berezansky & V. A. Tesko

Authors
  1. S. Albeverio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. M. Berezansky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. A. Tesko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 5, pp. 588–617, May, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Albeverio, S., Berezansky, Y.M. & Tesko, V.A. A generalization of an extended stochastic integral. Ukr Math J 59, 645–677 (2007). https://doi.org/10.1007/s11253-007-0044-x

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-007-0044-x

Keywords

Search

Navigation