Skip to main content

Double Lusin condition and Vitali convergence theorem for the Itô–McShane Integral

Abstract

In this paper, we formulate a version of Vitali convergence theorem for the Itô–McShane integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also characterize the integral using double Lusin condition.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Cabral, E., Lee, P.Y.: A fundamental theorem of calculus for the Kuzweil–Henstock integral in \({\mathbb{R}}^m\). Real Anal. Exch. 26, 867–876 (2000)

    Article  Google Scholar 

  2. 2.

    Chew, T.S., Toh, T.L., Tay, J.Y.: The non-uniform Riemann approach to Itô’s integral. Real Anal. Exch. 27, 495–514 (2001)

    Article  Google Scholar 

  3. 3.

    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  4. 4.

    Gawarecki, L., Mandrekar, V.: Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations. Springer, Berlin (2011)

    Book  Google Scholar 

  5. 5.

    Gordon, R.A.: The integrals of Lebesgue, Denjoy, Perron and Henstock. American Mathematical Society, Providence, Rhode Island (1994)

    Book  Google Scholar 

  6. 6.

    Henstock, R.: Lectures on the theory of integration. World Scientific, Singapore (1988)

    Book  Google Scholar 

  7. 7.

    Kurzweil, J.: Henstock–Kurzweil integration: its relation to topological vector spaces. World Scientific, Singapore (2000)

    Book  Google Scholar 

  8. 8.

    Kurzweil, J., Schwabik, S.: McShane equi-integrability and Vitali’s convergence theorem. Math. Bohem. 129, 141–157 (2004)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Labendia, M.: A Riemann-type definition of the Itô integral for the operator-valued stochastic process. Adv. Oper. Theory 4, 625–640 (2019)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Labendia, M., Arcede, J.: A descriptive definition of the Itô–Henstock integral for the operator-valued stochastic process. Adv. Oper. Theory 4, 406–418 (2019)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Labendia, M., De Lara-Tuprio, E., Teng, T.R.: Itô–Henstock integral and Itô’s formula for the operator-valued stochastic process. Math. Bohem. 143, 135–160 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lee, P.Y.: Lanzhou lectures on henstock integration. World Scientific, Singapore (1989)

    MATH  Google Scholar 

  13. 13.

    Lee, P.Y., Výborný, R.: The integral: an easy approach after Kurzwiel and Henstock. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  14. 14.

    Lee, T.Y.: Henstock–Kurzweil integration on euclidean spaces. World Scientific, Singapore (2011)

    Book  Google Scholar 

  15. 15.

    Lu, J.T., Lee, P.Y.: The primitives of Henstock integrable functions in Euclidean space. Bull. Lond. Math. Soc. 31, 173–180 (1999)

    MathSciNet  Article  Google Scholar 

  16. 16.

    McShane, E.J.: Stochastic calculus and stochastic models. Academic Press, Oval Road, London (1974)

    MATH  Google Scholar 

  17. 17.

    McShane, E.J.: Stochastic integrals and stochastic functional equations. SIAM J. Appl. Math. 17, 287–306 (1969)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Min, M.Z., Lee, P.Y., Chew, T.S.: Absolute integration using vitali covers. Real Anal. Exch. 18, 409–419 (1992)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Pop-Stojanovic, Z.R.: On McShane’s belated stochastic integral. SIAM J. Appl. Math. 22, 87–92 (1972)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Prévót, C., Röckner, M.: A concise course on stochastic partial differential equations. Springer, Berlin (2007)

    MATH  Google Scholar 

  21. 21.

    Reed, M., Simon, B.: Methods of modern mathematical physics I: functional analysis. Academic Press, London (1980)

    MATH  Google Scholar 

  22. 22.

    Royden, H., Fitzpatrick, P.: Real analysis, 4th edn. Prentice-Hall, New York (2007)

    MATH  Google Scholar 

  23. 23.

    Rulete, R., Labendia, M.: A descriptive definition of the backwards Itô–Henstock integral. Real Analysis Exchange (accepted)

  24. 24.

    Rulete, R., Labendia, M.: Double Lusin condition and convergence theorems for the backwards Itô–Henstock Integral. Real Anal. Exch. (accepted)

  25. 25.

    Toh, T.L., Chew, T.S.: On belated differentiation and a characterization of Henstock–Kurzweil-Itô integrable processes. Math. Bohem. 130, 63–72 (2005)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Toh, T.L., Chew, T.S.: On Itô–Kurzweil–Henstock integral and integration-by-part formula. Czechoslov. Math. J. 55, 653–663 (2005)

    Article  Google Scholar 

  27. 27.

    Toh, T.L., Chew, T.S.: On the Henstock–Fubini theorem for multiple stochastic integrals. Real Anal. Exch. 30, 295–310 (2004)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Toh, T.L., Chew, T.S.: The Riemann approach to stochastic integration using non-uniform meshes. J. Math. Anal. Appl. 280, 133–147 (2003)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mhelmar A. Labendia.

Additional information

Communicated by Un Cig Ji.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Labendia, M.A., Cagubcob, J.D.A. Double Lusin condition and Vitali convergence theorem for the Itô–McShane Integral. Adv. Oper. Theory 5, 453–473 (2020). https://doi.org/10.1007/s43036-019-00038-5

Download citation

Keywords

  • Belated McShane integral
  • Itô–McShane integral
  • \({{\mathcal {I}}}{{\mathcal {M}}}\)-equi-integrable
  • \({{\mathcal {I}}}{{\mathcal {M}}}\)-equi-AC
  • Q-Wiener process

Mathematics Subject Classification

  • 60H30
  • 60H05